Examples

$x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4$ , $x - 4$

Step 1

Divide $\frac{x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4}{x - 4}$ using synthetic division and check if the remainder is equal to $0$ . If the remainder is equal to $0$ , it means that $x - 4$ is a factor for $x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4$ . If the remainder is not equal to $0$ , it means that $x - 4$ is not a factor for $x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4$ .

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Step 1.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$4$	$1$	$- 2$	$- 10$	$7$	$4$

Step 1.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$4$	$1$	$- 2$	$- 10$	$7$	$4$

	$1$

Step 1.3

Multiply the newest entry in the result $(1)$ by the divisor $(4)$ and place the result of $(4)$ under the next term in the dividend $(- 2)$ .

$4$	$1$	$- 2$	$- 10$	$7$	$4$
		$4$
	$1$

Step 1.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$4$	$1$	$- 2$	$- 10$	$7$	$4$
		$4$
	$1$	$2$

Step 1.5

Multiply the newest entry in the result $(2)$ by the divisor $(4)$ and place the result of $(8)$ under the next term in the dividend $(- 10)$ .

$4$	$1$	$- 2$	$- 10$	$7$	$4$
		$4$	$8$
	$1$	$2$

Step 1.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$4$	$1$	$- 2$	$- 10$	$7$	$4$
		$4$	$8$
	$1$	$2$	$- 2$

Step 1.7

Multiply the newest entry in the result $(- 2)$ by the divisor $(4)$ and place the result of $(- 8)$ under the next term in the dividend $(7)$ .

$4$	$1$	$- 2$	$- 10$	$7$	$4$
		$4$	$8$	$- 8$
	$1$	$2$	$- 2$

Step 1.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$4$	$1$	$- 2$	$- 10$	$7$	$4$
		$4$	$8$	$- 8$
	$1$	$2$	$- 2$	$- 1$

Step 1.9

Multiply the newest entry in the result $(- 1)$ by the divisor $(4)$ and place the result of $(- 4)$ under the next term in the dividend $(4)$ .

$4$	$1$	$- 2$	$- 10$	$7$	$4$
		$4$	$8$	$- 8$	$- 4$
	$1$	$2$	$- 2$	$- 1$

Step 1.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$4$	$1$	$- 2$	$- 10$	$7$	$4$
		$4$	$8$	$- 8$	$- 4$
	$1$	$2$	$- 2$	$- 1$	$0$

Step 1.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 2 x^{2} + (- 2) x - 1$

Step 1.12

Simplify the quotient polynomial.

$x^{3} + 2 x^{2} - 2 x - 1$

Step 2

The remainder from dividing $\frac{x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4}{x - 4}$ is $0$ , which means that $x - 4$ is a factor for $x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4$ .

$x - 4$ is a factor for $x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4$

Step 3

Find all the possible roots for $x^{3} + 2 x^{2} - 2 x - 1$ .

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Step 3.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1$

$q = \pm 1$

Step 3.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1$

Step 4

Set up the next division to determine if $x - 1$ is a factor of the polynomial $x^{3} + 2 x^{2} - 2 x - 1$ .

$\frac{x^{3} + 2 x^{2} - 2 x - 1}{x - 1}$

Step 5

Divide the expression using synthetic division to determine if it is a factor of the polynomial. Since $x - 1$ divides evenly into $x^{3} + 2 x^{2} - 2 x - 1$ , $x - 1$ is a factor of the polynomial and there is a remaining polynomial of $x^{2} + 3 x + 1$ .

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Step 5.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$1$	$1$	$2$	$- 2$	$- 1$

Step 5.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$1$	$1$	$2$	$- 2$	$- 1$

	$1$

Step 5.3

Multiply the newest entry in the result $(1)$ by the divisor $(1)$ and place the result of $(1)$ under the next term in the dividend $(2)$ .

$1$	$1$	$2$	$- 2$	$- 1$
		$1$
	$1$

Step 5.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$1$	$2$	$- 2$	$- 1$
		$1$
	$1$	$3$

Step 5.5

Multiply the newest entry in the result $(3)$ by the divisor $(1)$ and place the result of $(3)$ under the next term in the dividend $(- 2)$ .

$1$	$1$	$2$	$- 2$	$- 1$
		$1$	$3$
	$1$	$3$

Step 5.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$1$	$2$	$- 2$	$- 1$
		$1$	$3$
	$1$	$3$	$1$

Step 5.7

Multiply the newest entry in the result $(1)$ by the divisor $(1)$ and place the result of $(1)$ under the next term in the dividend $(- 1)$ .

$1$	$1$	$2$	$- 2$	$- 1$
		$1$	$3$	$1$
	$1$	$3$	$1$

Step 5.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$1$	$2$	$- 2$	$- 1$
		$1$	$3$	$1$
	$1$	$3$	$1$	$0$

Step 5.9

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{2} + 3 x + 1$

Step 5.10

Simplify the quotient polynomial.

$x^{2} + 3 x + 1$

Step 6

Find all the possible roots for $x^{2} + 3 x + 1$ .

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Step 6.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1$

$q = \pm 1$

Step 6.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1$

Step 7

The final factor is the only factor left over from the synthetic division.

$x^{2} + 3 x + 1$

Step 8

The factored polynomial is $(x - 4) (x - 1) (x^{2} + 3 x + 1)$ .

$(x - 4) (x - 1) (x^{2} + 3 x + 1)$

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