Precalculus Examples

$x^{3} + x^{2} - 10 x + 8$ , $x - 2$

Step 1

Divide $\frac{x^{3} + x^{2} - 10 x + 8}{x - 2}$ using synthetic division and check if the remainder is equal to $0$ . If the remainder is equal to $0$ , it means that $x - 2$ is a factor for $x^{3} + x^{2} - 10 x + 8$ . If the remainder is not equal to $0$ , it means that $x - 2$ is not a factor for $x^{3} + x^{2} - 10 x + 8$ .

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Step 1.1

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

$2$	$1$	$1$	$- 10$	$8$

Step 1.2

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

$2$	$1$	$1$	$- 10$	$8$

	$1$

Step 1.3

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

$2$	$1$	$1$	$- 10$	$8$
		$2$
	$1$

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.

$2$	$1$	$1$	$- 10$	$8$
		$2$
	$1$	$3$

Step 1.5

Multiply the newest entry in the result $(3)$ by the divisor $(2)$ and place the result of $(6)$ under the next term in the dividend $(- 10)$ .

$2$	$1$	$1$	$- 10$	$8$
		$2$	$6$
	$1$	$3$

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.

$2$	$1$	$1$	$- 10$	$8$
		$2$	$6$
	$1$	$3$	$- 4$

Step 1.7

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

$2$	$1$	$1$	$- 10$	$8$
		$2$	$6$	$- 8$
	$1$	$3$	$- 4$

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.

$2$	$1$	$1$	$- 10$	$8$
		$2$	$6$	$- 8$
	$1$	$3$	$- 4$	$0$

Step 1.9

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

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

Step 1.10

Simplify the quotient polynomial.

$x^{2} + 3 x - 4$

Step 2

The remainder from dividing $\frac{x^{3} + x^{2} - 10 x + 8}{x - 2}$ is $0$ , which means that $x - 2$ is a factor for $x^{3} + x^{2} - 10 x + 8$ .

$x - 2$ is a factor for $x^{3} + x^{2} - 10 x + 8$

Step 3

Find all the possible roots for $x^{2} + 3 x - 4$ .

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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.

$p = \pm 1, \pm 2, \pm 4$

$q = \pm 1$

Step 3.2

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

$\pm 1, \pm 2, \pm 4$

Step 4

Set up the next division to determine if $x - 1$ is a factor of the polynomial $x^{2} + 3 x - 4$ .

$\frac{x^{2} + 3 x - 4}{x - 1}$

Step 5

Divide the expression using synthetic division to determine if it is a factor of the polynomial. Since $x - 1$ divides evenly into $x^{2} + 3 x - 4$ , $x - 1$ is a factor of the polynomial and there is a remaining polynomial of $x + 4$ .

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Step 5.1

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

$1$	$1$	$3$	$- 4$

Step 5.2

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

$1$	$1$	$3$	$- 4$

	$1$

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 $(3)$ .

$1$	$1$	$3$	$- 4$
		$1$
	$1$

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.

$1$	$1$	$3$	$- 4$
		$1$
	$1$	$4$

Step 5.5

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

$1$	$1$	$3$	$- 4$
		$1$	$4$
	$1$	$4$

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.

$1$	$1$	$3$	$- 4$
		$1$	$4$
	$1$	$4$	$0$

Step 5.7

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

$(1) x + 4$

Step 5.8

Simplify the quotient polynomial.

$x + 4$

Step 6

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

$x + 4$

Step 7

The factored polynomial is $(x - 2) (x - 1) (x + 4)$ .

$(x - 2) (x - 1) (x + 4)$