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 (*αx*–*β*), 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.

## 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 $latex x+1$.

##### Solution

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

**EXAMPLE**

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

##### Solution

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

**EXAMPLE**

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

##### Solution

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

**EXAMPLE**

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

##### Solution

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

**EXAMPLE**

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*.

##### Solution

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

**EXAMPLE**

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*.

##### Solution

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.

## See also

Interested in learning more about the remainder and factor theorems? You can take a look at these pages:

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