Precalculus Examples

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

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

x - 3

$x - 3$

Step 1

Divide

\frac{x^{3} - 3 x^{2} - 2 x + 6}{x - 3}

$\frac{x^{3} - 3 x^{2} - 2 x + 6}{x - 3}$ using synthetic division and check if the remainder is equal to

0

$0$ . If the remainder is equal to

0

$0$ , it means that

x - 3

$x - 3$ is a factor for

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

$x^{3} - 3 x^{2} - 2 x + 6$ . If the remainder is not equal to

0

$0$ , it means that

x - 3

$x - 3$ is not a factor for

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

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

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Step 1.1

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

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

Step 1.2

The first number in the dividend

(1)

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

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

	$1$ $1$

Step 1.3

Multiply the newest entry in the result

(1)

$(1)$ by the divisor

(3)

$(3)$ and place the result of

(3)

$(3)$ under the next term in the dividend

(- 3)

$(- 3)$ .

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

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.

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

Step 1.5

Multiply the newest entry in the result

(0)

$(0)$ by the divisor

(3)

$(3)$ and place the result of

(0)

$(0)$ under the next term in the dividend

(- 2)

$(- 2)$ .

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

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.

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

Step 1.7

Multiply the newest entry in the result

(- 2)

$(- 2)$ by the divisor

(3)

$(3)$ and place the result of

(- 6)

$(- 6)$ under the next term in the dividend

(6)

$(6)$ .

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

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.

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

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

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

Step 1.10

Simplify the quotient polynomial.

x^{2} - 2

$x^{2} - 2$

x^{2} - 2

$x^{2} - 2$

Step 2

The remainder from dividing

\frac{x^{3} - 3 x^{2} - 2 x + 6}{x - 3}

$\frac{x^{3} - 3 x^{2} - 2 x + 6}{x - 3}$ is

0

$0$ , which means that

x - 3

$x - 3$ is a factor for

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

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

x - 3

$x - 3$ is a factor for

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

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

Step 3

Find all the possible roots for

x^{2} - 2

$x^{2} - 2$ .

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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}

$\frac{p}{q}$ where

p

$p$ is a factor of the constant and

q

$q$ is a factor of the leading coefficient.

p = \pm 1, \pm 2

$p = \pm 1, \pm 2$

q = \pm 1

$q = \pm 1$

Step 3.2

Find every combination of

\pm \frac{p}{q}

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

\pm 1, \pm 2

$\pm 1, \pm 2$

\pm 1, \pm 2

$\pm 1, \pm 2$

Step 4

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

x^{2} - 2

$x^{2} - 2$

Step 5

The factored polynomial is

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

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

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

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

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