Remainder Theorem: Introduction
Just like we divide a number by another number, we can also divide one polynomial by another.
When we divide a polynomial by a linear polynomial, i.e., a polynomial of degree 1, we apply the remainder theorem to find the remainder.
Consider an example. The division

Similarly, if we divide a polynomial f(x) by a linear polynomial g(x) and get a remainder as r(x), we can express it as:

One way of finding the remainder is by long division of polynomials. This gives us the dividend and the quotient. Instead of this, we can use the theorem!
The remainder theorem of polynomials allows us to calculate the remainder of the division of a polynomial without carrying out the steps of long division.
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What Is the Remainder Theorem?
The remainder theorem states that when we divide a polynomial p
In simple words, if p
Using this theorem, the remainder of the division of any polynomial by a linear polynomial can be easily calculated.
Note:
- The degree of the remainder polynomial is always 1 less than the degree of the divisor polynomial. So, if any polynomial is divided by a linear polynomial (polynomial with degree
), the remainder is always constant (degree ). - We can say that
is the divisor of the polynomial P(x) if and only if P(a) . So, the theorem is applied to factorize polynomials.
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Remainder Theorem: Statement
The remainder theorem states that the remainder when a polynomial p(x) is divided by a linear polynomial
To find the remainder in a polynomial division, we can follow the given steps:
- Equate the linear polynomial or the divisor to 0 to find its “zero.”
So,
- Substitute
in the given polynomial p(x) to find the remainder.
Remainder
Example: Find the remainder when
Method 1: Long division

Remainder
Method 2: Remainder theorem
Here,
Thus,
Remainder
Remainder Theorem: Proof
By the division algorithm, we can write
Dividend
Let us assume that q(x) is the quotient and “r” is the remainder when a polynomial p(x) is divided by a linear polynomial
Applying this to polynomial division, we get:
If we substitute
Therefore, the remainder
How to Divide a Polynomial by a Non-Zero Polynomial?
The steps are as follows:
1. Arrange the polynomials (dividend and divisor) in decreasing order of their degree.
2. Divide the dividend polynomial’s first term by the divisor’s first term to obtain the first term of the quotient.
3. Multiply the divisor polynomial by the quotient’s first term.
3. Subtract the resulting product from the dividend to obtain the remainder.
4. This remainder is the dividend now, and the divisor will stay the same.
5. Repeat the process from the first step again until the degree of the new dividend is less than the degree of the divisor.
How to Use the Remainder Theorem of Polynomials?
Note the following important points when finding remainder using the remainder theorem.
- The remainder when p(x) is divided by
is p(a)
since
- The remainder when p(x) is divided by
is
since
- The remainder when p(x) is divided by
is
since
- The remainder when p(x) is divided by
is
since
Remainder Theorem examples:
Example 1: Find the remainder when you divide
Here,
Remainder
So
Substituting
So, the remainder
Example 2: Find the remainder when you divide
Put
Here,
Remainder
Remainder Theorem vs. Factor Theorem
Take a look at the following table to understand the difference between remainder theorem and factor theorem.
Remainder Theorem | Factor Theorem |
The remainder theorem states that the remainder when p(x) is divided by a linear polynomial of the form | The factor theorem states that |
Remainder theorem is used to find the remainder of the polynomial division only when the divisor polynomial is linear. | Factor theorem helps to decide if a linear polynomial is a factor of the given polynomial or not. |
Facts about Remainder Theorem
Here are some facts about the remainder theorem:
- The remainder polynomial is always 1 less degree than the degree of the divisor polynomial.
- Based on this, we can say that when a polynomial is divided by a linear polynomial (whose degree is 1), the remainder is a constant (whose degree is 0).
- The basis of the theorem is that we can find the zero of the divisor linear polynomial by setting it to 0. We say, if
, then . This is why the divisor polynomial plays an important role in the remainder theorem.
Conclusion
The remainder theorem is important because it helps us divide a polynomial and find the remainder easily. Instead of going through the steps of a long division, we can directly plug in the remainder theorem formula. All we have to do is substitute the right value and solve the polynomial!
Solved Examples
1. If $p(x) = x^{3} + 2x + 1$ is divided by
Solution: Applying the theorem, we know that the remainder is
Here,
So, the remainder
2. Find the remainder when
Solution:
Here,
$2x − 1 = 0 \Rightarrow a = \frac{1}{2}$
Remainder
3. Check if
Solution:
4. If the polynomial
Solution. By the theorem, we know that
So,
Therefore, the remainder is 19.
5. Find the remainder when you divide
Solution. We know that dividend
Remainder
Since the divisor is
So
Hence, the remainder
6. Find the remainder when you divide
Solution: p(x)
Remainder = p(a), where divisor is
So
Hence, the remainder
Practice Problems
Remainder Theorem
If p(x), a polynomial, is divided by , what is the remainder?
Applying the theorem, we know that the remainder r = p(a), where p(x) is a polynomial and
Here,
So the remainder
If p(x), a polynomial, is divided by , what is the remainder?
Applying the remainder theorem, we know that the remainder
Here, the divisor is
Therefore, the remainder is p(5).
Find the remainder if the is divided by .
Here,
Remainder
The remainder when p(x) is divided by is
The remainder when p(x) is divided by
Frequently Asked Questions
When do we use the factor theorem?
Factor theorem helps us to check if the linear polynomial is a factor of a given polynomial.
What is the origin of the remainder theorem?
The remainder theorem finds its origin in the work of Chinese mathematician Sun Zi.
What if we apply the remainder theorem and the remainder turns out to be 0?
If the remainder is 0, the given divisor
What is the simplest way to apply the remainder formula?
Where
Can the remainder formula be used for non-linear polynomials?
No, to apply the formula, the divisor has to be a linear polynomial and cannot be a non-linear polynomial.