{"id":29604,"date":"2023-05-30T09:03:29","date_gmt":"2023-05-30T09:03:29","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=29604"},"modified":"2023-11-15T17:52:57","modified_gmt":"2023-11-15T17:52:57","slug":"algebraic-identities-definition-factorization-proof-examples-faqs","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/algebraic-identities","title":{"rendered":"Algebraic Identities: Definition, Factorization, Proof, Examples, FAQs"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-287bf2b9-8aba-4273-9d37-ff3c3475ea40\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/algebraic-identities#0-what-are-algebraic-identities>What Are Algebraic Identities?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/algebraic-identities#4-how-to-prove-algebra-identities>How to Prove Algebra Identities<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/algebraic-identities#10-list-of-standard-algebraic-identities>List of Standard Algebraic Identities<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/algebraic-identities#14-solved-examples-on-algebraic-identities>Solved Examples on Algebraic Identities<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/algebraic-identities#15-practice-problems-on-algebraic-identities>Practice Problems on Algebraic Identities<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/algebraic-identities#16-frequently-asked-questions-on-algebraic-identities>Frequently Asked Questions on Algebraic Identities<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-what-are-algebraic-identities\">What Are Algebraic Identities?<\/h2>\n\n\n\n<p><strong>An algebraic identity is basically an equation in which L.H.S. equals R.H.S. for all values of the variables.<\/strong> An identity in math is an equation that holds true for all the values, even if you change the variables involved. For every value of the variables, an algebraic identity indicates that the left and right sides of the equation are identical.<\/p>\n\n\n\n<p>Algebraic identities are equations that hold true for all values of variables.<\/p>\n\n\n\n<p>In mathematical identities, the values on the left and right sides of the equation are exactly the same. We use algebraic identities as a set of formulas that help us in simplifying and solving algebraic equations.&nbsp;<\/p>\n\n\n\n<p>Consider an example.<\/p>\n\n\n\n<p>To expand the algebraic expression $(x + 1)^2$, you will have to multiply $(x + 1)$ with itself. It will surely be lengthy and time consuming.<br>However, if you use the algebraic identity $(a + b)^2 = a^2 + 2ab + b^2$<strong>, <\/strong>you will simply have to substitute the values and your job is done!<\/p>\n\n\n\n<div id=\"recommended-games-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Games<\/h4><div class=\"recommended-games-container-slides\"><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/evaluate-algebraic-expressions-with-one-operation\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/algebra_evaluate_algebric_exp_1_pt.png\" alt=\"Evaluate Algebraic Expressions with One Operation Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Evaluate Algebraic Expressions with One Operation Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/evaluate-algebraic-expressions-with-two-operations\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/algebra_evaluate_algebric_exp_2_pt.png\" alt=\"Evaluate Algebraic Expressions with Two Operations Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Evaluate Algebraic Expressions with Two Operations Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/form-algebraic-expressions\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/algebra_choose_exp_1_pt.png\" alt=\"Form Algebraic Expressions Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Form Algebraic Expressions Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/games\">More Games<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-games-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".game-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading\" id=\"1-algebraic-identity-definition\">Algebraic Identity Definition<\/h2>\n\n\n\n<p><strong>An important set of mathematical formulas or equations where the value of the L.H.S. of the equation is equal to the value of the R.H.S. of the equation.&nbsp;<\/strong><\/p>\n\n\n\n<p>Algebraic identities simplify algebraic <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/number-sense\/expression\">expressions<\/a> and calculations.&nbsp;<\/p>\n\n\n\n<p>Here are examples of common algebraic identities:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$(a + b)^2 = a^2 + 2ab + b^2$<\/li>\n\n\n\n<li>$(a \\;\u2212\\; b)^2 = a^2 \\;\u2212\\; 2ab + b^2$<\/li>\n\n\n\n<li>$a^2 \\;\u2212\\; b^2 = (a + b)(a \\;\u2212\\; b)$<\/li>\n\n\n\n<li>$(x + a)(x + b) = x^2 + x(a + b) + ab$<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"2-algebraic-identities-with-two-variables\">Algebraic Identities with Two Variables<\/h2>\n\n\n\n<p>The algebraic identities with two variables are known as two variable identities. Some basic two-variable algebraic identities are as follows:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$(a + b)^2 = a^2 + 2ab + b^2$<\/li>\n\n\n\n<li>$(a \\;\u2212\\; b)^2 = a^2 \\;\u2212\\; 2ab + b^2$<\/li>\n\n\n\n<li>$a^2 \\;\u2212\\; b^2 = (a + b)(a \\;\u2212\\; b)$<\/li>\n\n\n\n<li>$(a + b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3$<\/li>\n\n\n\n<li>$(a \\;\u2212 \\;b)^3 = a^3 \\;\u2212\\; 3a^2b + 3ab^2 \\;\u2212\\; b^3$<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-algebraic-identities-with-three-variables-\">Algebraic Identities with Three Variables&nbsp;<\/h2>\n\n\n\n<p>Some of the most important algebraic identities that use three variables are as follows:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac$<\/li>\n\n\n\n<li>$a^3 + b^3 + c^3 \\;\u2212\\; 3abc = (a + b + c)(a^2 + b^2 + c^2 \\;\u2212\\; ab \\;\u2212\\; ca \\;\u2212\\; bc)$<\/li>\n<\/ul>\n\n\n\n<p>Two identities can be derived from the above two identities.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$a^2 + b^2 + c^2 = (a + b + c)^2 \\;\u2212\\; 2(ab + bc + ac)$<\/li>\n\n\n\n<li>$(a + b)(b + c)(c + a) = (a + b + c)(ab + ac + bc) \u2212 2abc$<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-how-to-prove-algebra-identities\">How to Prove Algebra Identities<\/h2>\n\n\n\n<p>Algebraic identities can be proved by verifying if the L.H.S. (left-hand side) is equal to R.H.S. (right-hand side). Here, you have to solve and simplify the expression on one side (or both sides) using simple arithmetic <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/addition\/operation\">operations<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-proof-of-algebra-identities\">Proof of Algebra Identities<\/h2>\n\n\n\n<p>Let&#8217;s look at the proof of some fundamental algebraic identities.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-proof-of-a-b-2-a-2-2ab-b-2-\">Proof of (a + b)<sup>2<\/sup> = a<sup>2<\/sup> + 2ab + b<sup>2<\/sup><\/h2>\n\n\n\n<p>This formula is used to find the square of a binomial involving <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/addition\/addition\">addition<\/a>.<\/p>\n\n\n\n<p>L.H.S. $= (a + b)^2$<\/p>\n\n\n\n<p>L.H.S. $= (a + b) \\times (a + b)$<\/p>\n\n\n\n<p>L.H.S. $= a^2 + ab + ba + b^2$<\/p>\n\n\n\n<p>L.H.S. $= a^2 + 2ab + b^2$<\/p>\n\n\n\n<p>L.H.S. $=$ R.H.S.<\/p>\n\n\n\n<p>Hence proved.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"354\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/visual-proof-of-a-b2-a2-2ab-b2.webp\" alt=\"visual proof of (a + b)2 = a2 + 2ab + b2\" class=\"wp-image-35955\" title=\"visual proof of (a + b)2 = a2 + 2ab + b2\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/visual-proof-of-a-b2-a2-2ab-b2.webp 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/visual-proof-of-a-b2-a2-2ab-b2-300x171.webp 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"7-proof-of-a-%E2%88%92-b-2-a-2-%E2%88%92-2ab-b-2-\">Proof of (a \u2212 b)<sup>2<\/sup> = a<sup>2<\/sup> \u2212 2ab + b<sup>2<\/sup><\/h2>\n\n\n\n<p>This identity helps to expand the square of the difference between the two variables.<\/p>\n\n\n\n<p>L.H.S. $= (a \\;\u2212\\; b)^2$<\/p>\n\n\n\n<p>L.H.S. $= (a \\;\u2212\\; b) \\times (a \\;\u2212\\; b)$<\/p>\n\n\n\n<p>L.H.S. $= a^2 \\;\u2212\\; ab \\;\u2212\\; ba + b^2$<\/p>\n\n\n\n<p>L.H.S. $= a^2 \\;\u2212\\; 2ab + b^2$<\/p>\n\n\n\n<p>L.H.S. $=$ R.H.S.<\/p>\n\n\n\n<p>Hence proved.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"412\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/visual-proof-of-a-\u2212-b2-a2-\u2212-2ab-b2.webp\" alt=\"visual proof of (a \u2212 b)2 = a2 \u2212 2ab + b2\" class=\"wp-image-35956\" title=\"visual proof of (a \u2212 b)2 = a2 \u2212 2ab + b2\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/visual-proof-of-a-\u2212-b2-a2-\u2212-2ab-b2.webp 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/visual-proof-of-a-\u2212-b2-a2-\u2212-2ab-b2-300x199.webp 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"8-proof-of-a-ba-b-a-2-b-2-\">Proof of (a + b)(a &#8211; b) = a<sup>2<\/sup> &#8211; b<sup>2<\/sup><\/h2>\n\n\n\n<p>This identity helps to find the difference between two squares without actually calculating the individual square numbers.<\/p>\n\n\n\n<p>L.H.S. $= (a + b)(a \\;\u2212\\; b)$<\/p>\n\n\n\n<p>L.H.S. $= a^2 \\;\u2212\\; ab + ba \\;\u2212\\; b^2$<\/p>\n\n\n\n<p>L.H.S. $= a^2 \\;\u2212 \\;b^2$<\/p>\n\n\n\n<p>L.H.S. $=$ R.H.S.<\/p>\n\n\n\n<p>Hence proved.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"419\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/visual-proof-of-a-b2-a2-2ab-b2-.webp\" alt=\"visual proof of (a + b)2 = a2 + 2ab + b2\" class=\"wp-image-35957\" title=\"visual proof of (a + b)2 = a2 + 2ab + b2\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/visual-proof-of-a-b2-a2-2ab-b2-.webp 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/visual-proof-of-a-b2-a2-2ab-b2--300x203.webp 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9-proof-of-x-ax-b-x-2-xa-b-ab\">Proof of (x + a)(x + b) = x<sup>2<\/sup> + x(a + b) + ab<\/h2>\n\n\n\n<p>L.H.S. $= (x + a)(x + b)$<\/p>\n\n\n\n<p>L.H.S. $= x^2 + bx + ax + ab$<\/p>\n\n\n\n<p>L.H.S. $= x^2 + x(a + b) + ab$<\/p>\n\n\n\n<p>L.H.S. $=$ R.H.S.<\/p>\n\n\n\n<p>Hence proved.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"356\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/visual-proof-of-x-ax-b-x2-xa-b-ab.webp\" alt=\"visual proof of (x + a)(x + b) = x2 + x(a + b) + ab\" class=\"wp-image-35959\" title=\"visual proof of (x + a)(x + b) = x2 + x(a + b) + ab\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/visual-proof-of-x-ax-b-x2-xa-b-ab.webp 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/visual-proof-of-x-ax-b-x2-xa-b-ab-300x172.webp 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"10-list-of-standard-algebraic-identities\">List of Standard Algebraic Identities<\/h2>\n\n\n\n<figure class=\"wp-block-table wj-html-table\"><table class=\"wj-table-class\"><tbody><tr><td>Identity 1<\/td><td>$(a + b)^2 = a^2 + 2ab + b^2$<\/td><\/tr><tr><td>Identity 2<\/td><td>$(a \\;\u2013\\; b)^2 = a^2 \u2013 2ab + b^2$<\/td><\/tr><tr><td>Identity 3<\/td><td>$a^2 \\;\u2013\\; b^2 = (a + b)(a \\;\u2013\\; b)$<\/td><\/tr><tr><td>Identity 4<\/td><td>$(x + a)(x + b) = x^2 + (a + b)x + ab$<\/td><\/tr><tr><td>Identity 5<\/td><td>$(a + b)^3 = a^3 + b^3 + 3ab (a + b)$<\/td><\/tr><tr><td>Identity 6<\/td><td>$(a \\;\u2013\\; b)^3 = a^3 \\;\u2013\\; b^3 \\;\u2013\\; 3ab (a \\;\u2013\\; b)$<\/td><\/tr><tr><td>Identity 7<\/td><td>$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$<\/td><\/tr><tr><td>Identity 8<\/td><td>$a^3 + b^3 + c^3\\;\u2013\\; 3abc = (a + b + c)(a^2 + b^2 + c^2 \\;\u2013\\; ab \\;\u2013\\; bc \\;\u2013\\; ca)$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"11-factorization-identities-list\">Factorization Identities List<\/h2>\n\n\n\n<p>Some of the higher algebraic expressions, like $a^4 \\;\u2013\\; b^4,\\; a^6 \\;\u2013\\; b^6,\\; a^6 + b^6$, etc., can be easily factorized with a collection of algebraic identities that can be used to factor polynomials. The following are some common factorization identities.<\/p>\n\n\n\n<figure class=\"wp-block-table wj-custom-table\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\"><strong>Factorization Identities<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-left\" data-align=\"left\">$a^2 \\;\u2013\\; b^2 = (a + b)(a \\;\u2013\\; b)$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">$(x + a)(x + b) = x^2 + (a + b)x + ab$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">$a^3 \\;\u2013\\; b^3 = (a \\;\u2013\\; b)(a^2 + ab + b^2)$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">$a^3 + b^3 = (a + b)(a^2 \\;\u2013\\; ab + b^2)$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"12-facts-about-algebraic-identities\">Facts about Algebraic Identities<\/h2>\n\n\n\n<div style=\"color:#0369a1;background-color:#e0f2fe\" class=\"wp-block-roelmagdaleno-callout-block has-text-color has-background is-layout-flex wp-container-roelmagdaleno-callout-block-is-layout-8cf370e7 wp-block-roelmagdaleno-callout-block-is-layout-flex\"><div>\n<p>An algebraic identity in terms of \u201ca\u201d and \u201cb\u201d holds true for all the values of a and b.<br>The binomial theorem describes the algebraic expansion of powers of a binomial, $(x + y)^n$. It is also known as the \u201cbinomial identity.\u201d<\/p>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"13-conclusion\">Conclusion<\/h2>\n\n\n\n<p>In this article, we have learned about all algebraic identities, proofs, and related facts. Let\u2019s solve a few examples and practice problems based on the list of algebraic identities.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"14-solved-examples-on-algebraic-identities\">Solved Examples on Algebraic Identities<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Find the value of <\/strong>$195 \\times 205$<strong>.<\/strong><\/li>\n<\/ol>\n\n\n\n<p><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p>$195 \\times 205$ can be written as $(200 \\;\u2013\\; 5) \\times ( 200 + 5)$<\/p>\n\n\n\n<p>Applying $(a + b)(a \\;\u2013\\; b) = a^2 \\;\u2013\\; b^2$ with $a = 200$, and $b = 5$, we get<\/p>\n\n\n\n<p>$(200 \\;\u2013\\; 5) \\times ( 200 + 5) = 200^2 \\;-\\; 5^2$<\/p>\n\n\n\n<p>$= 40000 \\;\u2013\\; 25$<\/p>\n\n\n\n<p>$= 39975$<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li><strong>Using algebraic identities simplify <\/strong>$(5p \\;\u2013\\; 6q)^2 + (5p + 6q)^2$<\/li>\n<\/ol>\n\n\n\n<p><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p>The given expression is in the form of $(a \\;\u2013\\; b)^2 + (a + b)^2$<\/p>\n\n\n\n<p>So, we have to apply the following identities<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$(a + b)^2 = a^2 + 2ab + b^2$<\/li>\n\n\n\n<li>$(a \\;\u2013\\; b)^2 = a^2 \\;\u2013\\; 2ab + b^2$<\/li>\n<\/ul>\n\n\n\n<p>Here, $a = 5p$, and $b = 6q$<\/p>\n\n\n\n<p>$(5p \\;\u2013\\; 6q)^2 = (5p)^2 \\;\u2013\\; 2(5p)(6q) + (6q)^2 = 25p^2 \\;\u2013\\; 60pq + 36q^2$ \u2026 (i)<\/p>\n\n\n\n<p>$(5p + 6q)^2 = (5p)^2 + 2(5p)(6q) + (6q)^2 = 25p^2 + 60pq + 36q^2$ \u2026 (ii)<\/p>\n\n\n\n<p>Now adding equations (i) and (ii), we get<\/p>\n\n\n\n<p>&nbsp;&nbsp;$(5p \\;\u2013\\; 6q)^2 + (5p + 6q)^2 = 25p^2 \\;\u2013\\; 60pq + 36q^2 + 25p^2 + 60pq + 36q^2$<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$= 2(25p^2 + 36q^2)$<\/p>\n\n\n\n<p>$\\therefore (5p \\;\u2013\\; 6q)^2 + (5p + 6q)^2 = 2(25p^2 + 36q^2)$<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\">\n<li><strong>Using standard algebraic identities, determine the product of <\/strong>$(2k + 4)$<strong> and <\/strong>$(2k + 9)$<strong>.<\/strong><\/li>\n<\/ol>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>Given multiplication is $(2k + 4) \\times (2k + 9)$<\/p>\n\n\n\n<p>Apply $(x + a)(x + b) = x^2 + (a + b)x + ab$<\/p>\n\n\n\n<p>Here $x = 2k,\\; a = 4,$ and $b = 9$<\/p>\n\n\n\n<p>So, $(2k + 4) \u00d7 (2k + 9)&nbsp; = (2k)^2 + (4 + 9)2k + (4)(9)$<\/p>\n\n\n\n<p>$= 4k^2 + 26k + 36$<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"4\">\n<li><strong>Using standard identities simplify <\/strong>$(2x \\;\u2013\\; y)^3 + ( 2x + y )^3$<\/li>\n<\/ol>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>The given expression is in the form of $(a \\;\u2013\\; b)^3 + (a + b)^3$<\/p>\n\n\n\n<p>So, we have to apply the following identities<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$(a + b)^3 = a^3 + b^3 + 3ab (a + b)$<\/li>\n\n\n\n<li>$(a \\;\u2013\\; b)^3 = a^3 \\;\u2013\\; b^3 \\;\u2013\\; 3ab (a \\;\u2013\\; b)$<\/li>\n<\/ul>\n\n\n\n<p>Here, $a = 2x$, and $b = y$<\/p>\n\n\n\n<p>$(2x \\;\u2013\\; y)^3 = (2x)^3 \\;\u2013\\; y^3 \\;\u2013\\; 3(2x)(y) (2x \\;\u2013\\; y) = 8x^3 \\;\u2013\\; y^3 \\;\u2013\\; 6x^2 y + 6xy^2$ \u2026 (i)<\/p>\n\n\n\n<p>$(2x + y)^3 = (2x)^3 + y^3 + 3(2x)(y) (2x + y) = 8x^3 + y^3 + 6x^2y + 6xy^2$ \u2026 (i)<\/p>\n\n\n\n<p>Adding equations (i) and (ii), we get<\/p>\n\n\n\n<p>$(2x \\;\u2013\\; y)^3 + (2x + y)^3 = 8x^3 \\;\u2013\\; y^3 \\;\u2013\\; 6x^2 y + 6xy^2 + 8x^3 + y^3 + 6x^2 y + 6xy^2$<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$= 2(8x^3 + 6xy^2) = 4(4x^3 + 3xy^2)$<\/p>\n\n\n\n<p>$\\therefore (2x \\;\u2013\\; y)^3 + (2x + y)^3 = 4(4x^3 + 3xy^2)$<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"5\">\n<li><strong>Using factorization identities, factorize <\/strong>$(8x^3 + 27y^3)$<strong>.<\/strong><\/li>\n<\/ol>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>The given expression can be written as $(8x^3 + 27y^3) = (2x)^3 + (3y)^3$,&nbsp;<\/p>\n\n\n\n<p>which is in the form of $a^3 + b^3$<\/p>\n\n\n\n<p>So, for factorization we have to apply the identities $a^3 + b^3 = (a + b)(a^2 \\;\u2013\\; ab + b^2)$<\/p>\n\n\n\n<p>Here, $a = 2x$ and $b = 3y$<\/p>\n\n\n\n<p>Now<\/p>\n\n\n\n<p>$(8x^3 + 27y^3) = (2x)^3 + (3y)^3$&nbsp;<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$= (2x + 3y)((2x)^2 \\;\u2013\\; (2x)(3y) + (3y)^2)$<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$= (2x + 3y)(4x^2 \\;\u2013\\; 6xy + 9y^2)$<\/p>\n\n\n\n<p>Hence, $(8x^3 + 27y^3) = (2x + 3y)(4x^2 \\;\u2013\\; 6xy + 9y^2)$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"15-practice-problems-on-algebraic-identities\">Practice Problems on Algebraic Identities<\/h2>\n\n\n\n<div class=\"spq_wrapper\"><h2 style=\"display:none;\">Algebraic Identities: Definition, Factorization, Proof, Examples, FAQs<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">Which of the following expressions is equal to $(2x + 3)^2$?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$4x^2 + 9$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$4x^2\\;\u2013\\;12x + 9$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$4x^2 + 12x + 9$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$4x^2\\;\u2013\\;9$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $4x^2 + 12x + 9$<br\/>We know that $(a + b)^{2} = a^{2} + 2ab + b^{2}$<br>\r\n$(2x + 3)^{2} = (2x)^{2} + 2(2x)(3) + 3^{2} = 4x^{2} + 12x + 9$.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">Which of the following is equal to $(x\\;\u2212\\;7)(x + 7)$?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$x^2\\;\u2212\\;39$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$x\\;\u2212\\;49$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$(x\\;\u2212\\;49)^2$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$x^2\\;\u2212\\;49$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $x^2\\;\u2212\\;49$<br\/>We know that $(a + b)(a\\;\u2013\\;b) = a^2\\;\u2013\\;b^2$.<br>\r\n$(x\\;\u2212\\;7)(x + 7) = x^2\\;\u2212\\;49$.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">Which of the following is equal to $x^3\\;\u2212\\;y^3$?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$(x\\;\u2013\\;y)^3$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$(x + y)^3$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$(x\\;\u2013\\;y)(x^2 + xy + y^2)$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$(x + y)(x^2\\;\u2013\\;xy + y^2)$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $(x\\;\u2013\\;y)(x^2 + xy + y^2)$<br\/>We know that $a^3\\;\u2013\\;b^3 = (a\\;\u2013\\;b)(a^2 + ab + b^2)$<br>\r\n$x^3\\;\u2212\\;y^3 = (x\\;\u2013\\;y)(x^2 + xy + y^2)$.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">Which of the following is an identity?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$x^2 + 2x + 1 = 0$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$x(x + 1) = x^2 + x$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$x\\;\u2013\\;5 = 2x$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$(2x\\;\u2013\\;3)2 = 0$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $x(x + 1) = x^2 + x$<br\/>We observe that $x(x + 1) = x^2 + x$ that is L.H.S $=$ R.H.S.<br>\r\nHence, it is an identity.<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        \"educationalLevel\":  \"beginner\",\n        \"assesses\" : \"Attend this quiz & Test your knowledge.\",\n        \"educationalAlignment\": [\n              {\n                \"@type\": \"AlignmentObject\",\n                \"alignmentType\": \"educationalSubject\",\n                \"targetName\": \"Math\"\n              }] ,\n        \"name\": \"Algebraic Identities: Definition, Factorization, Proof, Examples, FAQs\",        \n        \"about\": {\n              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    \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$4x^2 + 9$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that $$(a + b)^{2} = a^{2} + 2ab + b^{2}$$<br>\r\n$$(2x + 3)^{2} = (2x)^{2} + 2(2x)(3) + 3^{2} = 4x^{2} + 12x + 9$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$4x^2\\\\;\u2013\\\\;12x + 9$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that $$(a + b)^{2} = a^{2} + 2ab + b^{2}$$<br>\r\n$$(2x + 3)^{2} = (2x)^{2} + 2(2x)(3) + 3^{2} = 4x^{2} + 12x + 9$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$4x^2\\\\;\u2013\\\\;9$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that $$(a + b)^{2} = a^{2} + 2ab + b^{2}$$<br>\r\n$$(2x + 3)^{2} = (2x)^{2} + 2(2x)(3) + 3^{2} = 4x^{2} + 12x + 9$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$4x^2 + 12x + 9$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"We know that $$(a + b)^{2} = a^{2} + 2ab + b^{2}$$<br>\r\n$$(2x + 3)^{2} = (2x)^{2} + 2(2x)(3) + 3^{2} = 4x^{2} + 12x + 9$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"We know that $$(a + b)^{2} = a^{2} + 2ab + b^{2}$$<br>\r\n$$(2x + 3)^{2} = (2x)^{2} + 2(2x)(3) + 3^{2} = 4x^{2} + 12x + 9$$.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Which of the following is equal to $$(x\\\\;\u2212\\\\;7)(x + 7)$$?\",\n                    \"text\": \"Which of the following is equal to $$(x\\\\;\u2212\\\\;7)(x + 7)$$?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"We know that $$(a + b)(a\\\\;\u2013\\\\;b) = a^2\\\\;\u2013\\\\;b^2$$.<br>\r\n$$(x\\\\;\u2212\\\\;7)(x + 7) = x^2\\\\;\u2212\\\\;49$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$x^2\\\\;\u2212\\\\;39$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that $$(a + b)(a\\\\;\u2013\\\\;b) = a^2\\\\;\u2013\\\\;b^2$$.<br>\r\n$$(x\\\\;\u2212\\\\;7)(x + 7) = x^2\\\\;\u2212\\\\;49$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$x\\\\;\u2212\\\\;49$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that $$(a + b)(a\\\\;\u2013\\\\;b) = a^2\\\\;\u2013\\\\;b^2$$.<br>\r\n$$(x\\\\;\u2212\\\\;7)(x + 7) = x^2\\\\;\u2212\\\\;49$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$(x\\\\;\u2212\\\\;49)^2$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that $$(a + b)(a\\\\;\u2013\\\\;b) = a^2\\\\;\u2013\\\\;b^2$$.<br>\r\n$$(x\\\\;\u2212\\\\;7)(x + 7) = x^2\\\\;\u2212\\\\;49$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$x^2\\\\;\u2212\\\\;49$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"We know that $$(a + b)(a\\\\;\u2013\\\\;b) = a^2\\\\;\u2013\\\\;b^2$$.<br>\r\n$$(x\\\\;\u2212\\\\;7)(x + 7) = x^2\\\\;\u2212\\\\;49$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"We know that $$(a + b)(a\\\\;\u2013\\\\;b) = a^2\\\\;\u2013\\\\;b^2$$.<br>\r\n$$(x\\\\;\u2212\\\\;7)(x + 7) = x^2\\\\;\u2212\\\\;49$$.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Which of the following is equal to $$x^3\\\\;\u2212\\\\;y^3$$?\",\n                    \"text\": \"Which of the following is equal to $$x^3\\\\;\u2212\\\\;y^3$$?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"We know that $$a^3\\\\;\u2013\\\\;b^3 = (a\\\\;\u2013\\\\;b)(a^2 + ab + b^2)$$<br>\r\n$$x^3\\\\;\u2212\\\\;y^3 = (x\\\\;\u2013\\\\;y)(x^2 + xy + y^2)$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$(x\\\\;\u2013\\\\;y)^3$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that $$a^3\\\\;\u2013\\\\;b^3 = (a\\\\;\u2013\\\\;b)(a^2 + ab + b^2)$$<br>\r\n$$x^3\\\\;\u2212\\\\;y^3 = (x\\\\;\u2013\\\\;y)(x^2 + xy + y^2)$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$(x + y)^3$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that $$a^3\\\\;\u2013\\\\;b^3 = (a\\\\;\u2013\\\\;b)(a^2 + ab + b^2)$$<br>\r\n$$x^3\\\\;\u2212\\\\;y^3 = (x\\\\;\u2013\\\\;y)(x^2 + xy + y^2)$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$(x + y)(x^2\\\\;\u2013\\\\;xy + y^2)$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We know that $$a^3\\\\;\u2013\\\\;b^3 = (a\\\\;\u2013\\\\;b)(a^2 + ab + b^2)$$<br>\r\n$$x^3\\\\;\u2212\\\\;y^3 = (x\\\\;\u2013\\\\;y)(x^2 + xy + y^2)$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$(x\\\\;\u2013\\\\;y)(x^2 + xy + y^2)$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"We know that $$a^3\\\\;\u2013\\\\;b^3 = (a\\\\;\u2013\\\\;b)(a^2 + ab + b^2)$$<br>\r\n$$x^3\\\\;\u2212\\\\;y^3 = (x\\\\;\u2013\\\\;y)(x^2 + xy + y^2)$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"We know that $$a^3\\\\;\u2013\\\\;b^3 = (a\\\\;\u2013\\\\;b)(a^2 + ab + b^2)$$<br>\r\n$$x^3\\\\;\u2212\\\\;y^3 = (x\\\\;\u2013\\\\;y)(x^2 + xy + y^2)$$.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Which of the following is an identity?\",\n                    \"text\": \"Which of the following is an identity?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"We observe that $$x(x + 1) = x^2 + x$$ that is L.H.S $$=$$ R.H.S.<br>\r\nHence, it is an identity.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$x^2 + 2x + 1 = 0$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We observe that $$x(x + 1) = x^2 + x$$ that is L.H.S $$=$$ R.H.S.<br>\r\nHence, it is an identity.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$x\\\\;\u2013\\\\;5 = 2x$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We observe that $$x(x + 1) = x^2 + x$$ that is L.H.S $$=$$ R.H.S.<br>\r\nHence, it is an identity.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$(2x\\\\;\u2013\\\\;3)2 = 0$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"We observe that $$x(x + 1) = x^2 + x$$ that is L.H.S $$=$$ R.H.S.<br>\r\nHence, it is an identity.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$x(x + 1) = x^2 + x$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"We observe that $$x(x + 1) = x^2 + x$$ that is L.H.S $$=$$ R.H.S.<br>\r\nHence, it is an identity.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"We observe that $$x(x + 1) = x^2 + x$$ that is L.H.S $$=$$ R.H.S.<br>\r\nHence, it is an identity.\"\n                      }\n                    } \n\n                    }]}<\/script>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"16-frequently-asked-questions-on-algebraic-identities\">Frequently Asked Questions on Algebraic Identities<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\"><strong>What are the three variable identities?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\">\n\n<p>The identities in <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/algebra\/algebra\">algebra<\/a> with three variables are known as three-variable identities. Some basic three-variable algebraic identities are as follows:<\/p>\n\n\n\n<p>(i) (a + b + c)<sup>2<\/sup> = a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> + 2ab + 2bc + 2ac<\/p>\n\n\n\n<p>(ii) a<sup>3<\/sup> + b<sup>3<\/sup> + c<sup>3<\/sup> \u2212 3abc = (a + b + c)(a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> \u2212 ab \u2212 ca \u2212 bc)<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\"><strong>What are the factorization identities?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\">\n\n<p>Some common factorization identities are&nbsp;<\/p>\n\n\n\n<p>a<sup>2<\/sup> \u2013 b<sup>2<\/sup> = (a + b)(a \u2013 b), (x + a)(x + b) = x<sup>2<\/sup> + (a + b)x + ab<\/p>\n\n\n\n<p>a<sup>3<\/sup> \u2013 b<sup>3<\/sup> = (a \u2013 b)(a<sup>2<\/sup> + ab + b<sup>2<\/sup>)a<sup>3<\/sup> + b<sup>3<\/sup> = (a + b)(a<sup>2<\/sup> \u2013 ab + b<sup>2<\/sup>), etc.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\"><strong>How many basic algebraic identities can be found in math?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\">\n\n<p>There are four basic algebraic identities in math:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(a + b)<sup>2<\/sup> = a<sup>2<\/sup> + 2ab + b<sup>2<\/sup><\/li>\n\n\n\n<li>(a \u2013 b)<sup>2<\/sup> = a<sup>2<\/sup> \u2013 2ab + b<sup>2<\/sup><\/li>\n\n\n\n<li>a<sup>2<\/sup> \u2013 b<sup>2<\/sup> = (a + b)(a \u2013 b)<\/li>\n\n\n\n<li>(x + a)(x + b) = x<sup>2<\/sup> + (a + b)x + ab<\/li>\n<\/ul>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-3-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\"><strong>What is the difference between identities and expressions in algebra?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-3-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\">\n\n<p>An algebraic identity is an equation. It holds true for all the values of the variable.&nbsp;<\/p>\n\n\n\n<p>Algebraic identity: 7y + 2 = 16<\/p>\n\n\n\n<p>An algebraic expression is formed using terms involving <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/algebra\/variable\">variables<\/a> and <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/constant\">constants<\/a>; there is no <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/counting-and-comparison\/equal-sign\">equal sign<\/a>.<\/p>\n\n\n\n<p>Algebraic expression: 7y + 2<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-4-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\"><strong>What is the use of algebraic identities?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-4-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\">\n\n<p>The use of algebraic identities simplifies mathematics calculations. When it comes to resolving quadratic and higher-power equations, algebraic identities play a crucial role. Also, products of numbers and other quantities can be calculated using algebraic identities.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-5-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\"><strong>What is the algebraic identity for finding the sum of squares of two numbers?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-5-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\">\n\n<p>a2+b2= (a+b)2-2ab<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-6-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\"><strong>What are the 13 algebraic identities?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-6-fc9fa5ff-8c02-4266-9c35-04bb1eb3069f\">\n\n<ul class=\"wp-block-list\">\n<li>(a + b)<sup>2<\/sup>= a<sup>2 <\/sup>+ 2ab + b<sup>2<\/sup><\/li>\n\n\n\n<li>(a &#8211; b)<sup>2 <\/sup>= a<sup>2 <\/sup>&#8211; 2ab + b<sup>2<\/sup><\/li>\n\n\n\n<li>a<sup>2<\/sup> \u2013 b<sup>2<\/sup> = (a + b)(a \u2013 b)<\/li>\n\n\n\n<li>a<sup>3<\/sup> \u2013 b<sup>3<\/sup> = (a \u2013 b)(a<sup>2<\/sup> + ab + b<sup>2<\/sup>)<\/li>\n\n\n\n<li>a<sup>3<\/sup> + b<sup>3<\/sup> = (a + b)(a<sup>2<\/sup> \u2013 ab + b<sup>2<\/sup>)<\/li>\n\n\n\n<li>(a + b + c)<sup>2<\/sup> = a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> + 2ab + 2bc + 2ac<\/li>\n\n\n\n<li>a<sup>3<\/sup> + b<sup>3<\/sup> + c<sup>3<\/sup> \u2212 3abc = (a + b + c)(a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> \u2212 ab \u2212 ca \u2212 bc)<\/li>\n\n\n\n<li>(a \u2212 b \u2212 c)<sup>2<\/sup> = a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2 <\/sup>\u2212 2ab + 2bc + 2ac<\/li>\n\n\n\n<li>(a + b + c)<sup>2<\/sup> = a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> + 2ab + 2bc + 2ac<\/li>\n\n\n\n<li>(a + b \u2013 c)<sup>2<\/sup> = a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> + 2ab \u2013 2bc \u2013 2ac<\/li>\n\n\n\n<li>(a \u2013 b + c)<sup>2<\/sup> = a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> \u2013 2ab \u2013 2bc + 2ac<\/li>\n\n\n\n<li>(\u2013a + b + c)<sup>2<\/sup> = a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> \u2013 2ab + 2bc \u2013 2ac<\/li>\n\n\n\n<li>(a + b)<sup>3<\/sup> = a<sup>3<\/sup> + 3a<sup>2<\/sup>b + 3ab<sup>2<\/sup> + b<sup>3<\/sup><\/li>\n\n\n\n<li>(a \u2013 b)<sup>3<\/sup> = a<sup>3<\/sup> \u2013&nbsp; 3a<sup>2<\/sup>b + 3ab<sup>2<\/sup> \u2013 b<sup>3<\/sup><\/li>\n<\/ul>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Are Algebraic Identities? An algebraic identity is basically an equation in which L.H.S. equals R.H.S. for all values of the variables. An identity in math is an equation that holds true for all the values, even if you change the variables involved. For every value of the variables, an algebraic identity indicates that the &#8230; <a title=\"Algebraic Identities: Definition, Factorization, Proof, Examples, FAQs\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/algebraic-identities\" aria-label=\"More on Algebraic Identities: Definition, Factorization, Proof, Examples, FAQs\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-29604","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/29604","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=29604"}],"version-history":[{"count":16,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/29604\/revisions"}],"predecessor-version":[{"id":35960,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/29604\/revisions\/35960"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=29604"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=29604"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=29604"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}