Remainder Theorem – Formula and Proof

The remainder theorem is an algebraic theorem that allows us to determine what the remainder will be if we divide a polynomial by a given factor. The Remainder Theorem tells us that when we divide a polynomial f(x) by (αxβ), the remainder is f(β/α).

Here, we will learn everything related to the remainder theorem. We will look at its formula and will learn how to prove the theorem. In addition, we will solve some practice problems.

ALGEBRA
Formula for the remainder theorem

Relevant for

Learning about the remainder theorem with examples.

See theorem

ALGEBRA
Formula for the remainder theorem

Relevant for

Learning about the remainder theorem with examples.

See theorem

Statement and proof of the remainder theorem

The remainder theorem tells us that when we divide a polynomial $latex f(x)$ by $latex (\alpha x-\beta)$, the remainder is $latex f\left(\frac{\beta}{\alpha }\right)$.

Remainder Theorem Proof

We can prove the Remainder Theorem by writing as follows:

$$f(x)=(\alpha x-\beta)(\text{Quotient})+(\text{Remainder})$$

Now, we can use the value $latex x=\frac{\beta}{\alpha}$, to get the following:

$$f\left(\frac{\beta}{\alpha}\right)=\left[ \alpha \left(\frac{\beta}{\alpha}\right)-\beta\right](\text{Quotient})+(\text{Remainder})$$

$$=\left[ \beta-\beta\right](\text{Quotient})+(\text{Remainder})$$

$$=\left[ 0 \right](\text{Quotient})+(\text{Remainder})$$

$$=\text{Remainder}$$

Which shows that the Remainder Theorem is true.


Remainder theorem – Examples with answers

The following examples apply the Remainder Theorem to find the solution. Each example has a detailed solution, but try to solve the problems yourself before looking at the answer.

EXAMPLE 1

Find the remainder when the polynomial $latex x^3+5x^2-17x-21$ is divided by $latex x+1$.

The Remainder Theorem tells us that the remainder of the division is $latex f\left(\frac{\beta}{\alpha}\right)$. In this case, we have $latex f\left(\frac{\beta}{\alpha}\right)=f(-1)$. Thus, we have:

$$f(-1)=(-1)^3+5(-1)^2-17(-1)-21$$

$latex =-1+5+17-21$

$latex =0$

This means that the remainder is 0. That is, the division is exact.

EXAMPLE 2

If we now divide the polynomial $latex x^3+5x^2-17x-21$ by $latex x-4$, what is the remainder?

By the remainder theorem, we know that the remainder when $latex f(x)$ is divided by $latex x-4$ is $latex f(4)$. Therefore, we have:

$$f(4)=(4)^3+5(4)^2-17(4)-21$$

$latex =64+80-68-21$

$latex =55$

The remainder of the division is 55.

EXAMPLE 3

Find the remainder when the polynomial $latex x^3+5x^2-17x-21$ is divided by $latex 2x+1$

The remainder theorem tells us that the remainder when $latex f(x)$ is divided by $latex 2x+1$ is $latex f\left(-\frac{1}{2}\right)$. Therefore, we have:

$$f\left(-\frac{1}{2}\right)=\left(-\frac{1}{2}\right)^3+5\left(-\frac{1}{2}\right)^2-17\left(-\frac{1}{2}\right)-21$$

$latex =-\frac{1}{8}+\frac{5}{4}+\frac{17}{2}-21$

$latex =-\frac{91}{8}$

The remainder of the division is $latex -\frac{91}{8}$.

EXAMPLE 4

Find the remainder when the polynomial $latex 6x^3-2x^2+5x-4$ is divided by x.

When we compare x to $latex (\alpha x-\beta)$, we see that $latex \beta$ is equal to 0. Therefore, we use $latex f\left(\frac{\beta}{\alpha}\right) =f(0)$ to find the remainder.

$$f(0)=6(0)^3-2(0)^2+5(0)-4$$

$latex =0-0+0-4$

$latex =-4$

The remainder of the division is -4.

EXAMPLE 5

The polynomial $latex 3x^3+bx^2-7x+5$ gives us a remainder of 17 when divided by $latex x+3$. Find the value of b.

Using the remainder theorem, we know that $latex f(-3)=17$. Therefore, we have:

$$3(-3)^3+b(-3)^2-7(-3)+5=17$$

$latex -81+9b+21+5=17$

$latex 9b=72$

$latex b=8$

The value of b is 8.

EXAMPLE 6

When the polynomial $latex f(x)=x^3+ax^2+bx+2$ is divided by $latex x-1$, the remainder is 4 and when it is divided by $latex x+2$, the remainder is also 4. Find the values of a and b.

Using the Remainder Theorem, we know that $latex f(1)=4$. Therefore, we have:

$$(1)^3+a(1)^2+b(1)+2=4$$

$latex 1+a+b+2=4$

$latex a+b=1$

Now, we use the remainder theorem with $latex f(-2)=4$ and we have:

$$(-2)^3+a(-2)^2+b(-2)+2=4$$

$latex -8+4a-2b+2=4$

$latex 4a-2b=10$

Dividing the last equation by 2, we have:

$latex 2a-b=5$

Forming a system of equations with the two equations found and solving, we get $latex a=2$ and $latex b=-1$.


Remainder theorem – Practice problems

Use the Remainder Theorem to solve the following practice problems. You can use the examples with answers shown above as a guide.

Find the remainder when the polynomial $latex 6x^2+5x-1$ is divided by $latex x-1$.

Choose an answer






What is the remainder when $latex 3x^3+2x-4$ is divided by $latex x-2$?

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Find the remainder of dividing $latex 3x^2+6x-8$ by $latex x+3$.

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What is the remainder when we divide $latex 2x^3+4x^2-6x+5$ by $latex x-1$?

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When we divide $latex 2x^3-3x^2+ax-5$ by $latex x-2$, the remainder is 7. Find the value of a.

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