Precalculus Examples

$x^{3} - 3 x^{2} - 2 x + 6$ , $x - 3$

Step 1

Divide $\frac{x^{3} - 3 x^{2} - 2 x + 6}{x - 3}$ using synthetic division and check if the remainder is equal to $0$ . If the remainder is equal to $0$ , it means that $x - 3$ is a factor for $x^{3} - 3 x^{2} - 2 x + 6$ . If the remainder is not equal to $0$ , it means that $x - 3$ is not a factor for $x^{3} - 3 x^{2} - 2 x + 6$ .

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Step 1.1

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

$3$	$1$	$- 3$	$- 2$	$6$

Step 1.2

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

$3$	$1$	$- 3$	$- 2$	$6$

	$1$

Step 1.3

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

$3$	$1$	$- 3$	$- 2$	$6$
		$3$
	$1$

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.

$3$	$1$	$- 3$	$- 2$	$6$
		$3$
	$1$	$0$

Step 1.5

Multiply the newest entry in the result $(0)$ by the divisor $(3)$ and place the result of $(0)$ under the next term in the dividend $(- 2)$ .

$3$	$1$	$- 3$	$- 2$	$6$
		$3$	$0$
	$1$	$0$

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.

$3$	$1$	$- 3$	$- 2$	$6$
		$3$	$0$
	$1$	$0$	$- 2$

Step 1.7

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

$3$	$1$	$- 3$	$- 2$	$6$
		$3$	$0$	$- 6$
	$1$	$0$	$- 2$

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.

$3$	$1$	$- 3$	$- 2$	$6$
		$3$	$0$	$- 6$
	$1$	$0$	$- 2$	$0$

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.

$1 x^{2} + 0 x - 2$

Step 1.10

Simplify the quotient polynomial.

$x^{2} - 2$

Step 2

The remainder from dividing $\frac{x^{3} - 3 x^{2} - 2 x + 6}{x - 3}$ is $0$ , which means that $x - 3$ is a factor for $x^{3} - 3 x^{2} - 2 x + 6$ .

$x - 3$ is a factor for $x^{3} - 3 x^{2} - 2 x + 6$

Step 3

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

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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.

$p = \pm 1, \pm 2$

$q = \pm 1$

Step 3.2

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

$\pm 1, \pm 2$

Step 4

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

$x^{2} - 2$

Step 5

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

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

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