Precalculus Examples

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

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

x - 1

$x - 1$

Step 1

Divide

\frac{x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4}{x - 1}

$\frac{x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4}{x - 1}$ 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 - 1

$x - 1$ is a factor for

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

$x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4$ . If the remainder is not equal to

0

$0$ , it means that

x - 1

$x - 1$ is not a factor for

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

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

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Step 1.1

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

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

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).

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

	$1$ $1$

Step 1.3

Multiply the newest entry in the result

(1)

$(1)$ by the divisor

(1)

$(1)$ and place the result of

(1)

$(1)$ under the next term in the dividend

(- 2)

$(- 2)$ .

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

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.

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

Step 1.5

Multiply the newest entry in the result

(- 1)

$(- 1)$ by the divisor

(1)

$(1)$ and place the result of

(- 1)

$(- 1)$ under the next term in the dividend

(- 10)

$(- 10)$ .

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

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.

$1$ $1$	$1$ $1$	$- 2$ $- 2$	$- 10$ $- 10$	$7$ $7$	$4$ $4$
		$1$ $1$	$- 1$ $- 1$
	$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$

Step 1.7

Multiply the newest entry in the result

(- 11)

$(- 11)$ by the divisor

(1)

$(1)$ and place the result of

(- 11)

$(- 11)$ under the next term in the dividend

(7)

$(7)$ .

$1$ $1$	$1$ $1$	$- 2$ $- 2$	$- 10$ $- 10$	$7$ $7$	$4$ $4$
		$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$
	$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$

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.

$1$ $1$	$1$ $1$	$- 2$ $- 2$	$- 10$ $- 10$	$7$ $7$	$4$ $4$
		$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$
	$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$	$- 4$ $- 4$

Step 1.9

Multiply the newest entry in the result

(- 4)

$(- 4)$ by the divisor

(1)

$(1)$ and place the result of

(- 4)

$(- 4)$ under the next term in the dividend

(4)

$(4)$ .

$1$ $1$	$1$ $1$	$- 2$ $- 2$	$- 10$ $- 10$	$7$ $7$	$4$ $4$
		$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$	$- 4$ $- 4$
	$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$	$- 4$ $- 4$

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.

$1$ $1$	$1$ $1$	$- 2$ $- 2$	$- 10$ $- 10$	$7$ $7$	$4$ $4$
		$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$	$- 4$ $- 4$
	$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$	$- 4$ $- 4$	$0$ $0$

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} + - 1 x^{2} + (- 11) x - 4

$1 x^{3} + - 1 x^{2} + (- 11) x - 4$

Step 1.12

Simplify the quotient polynomial.

x^{3} - x^{2} - 11 x - 4

$x^{3} - x^{2} - 11 x - 4$

x^{3} - x^{2} - 11 x - 4

$x^{3} - x^{2} - 11 x - 4$

Step 2

The remainder from dividing

\frac{x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4}{x - 1}

$\frac{x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4}{x - 1}$ is

0

$0$ , which means that

x - 1

$x - 1$ is a factor for

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

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

x - 1

$x - 1$ is a factor for

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

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

Step 3

Find all the possible roots for

x^{3} - x^{2} - 11 x - 4

$x^{3} - x^{2} - 11 x - 4$ .

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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}

$\frac{p}{q}$ where

p

$p$ is a factor of the constant and

q

$q$ is a factor of the leading coefficient.

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

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

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 4

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

\pm 1, \pm 2, \pm 4

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

Step 4

Set up the next division to determine if

x - 4

$x - 4$ is a factor of the polynomial

x^{3} - x^{2} - 11 x - 4

$x^{3} - x^{2} - 11 x - 4$ .

\frac{x^{3} - x^{2} - 11 x - 4}{x - 4}

$\frac{x^{3} - x^{2} - 11 x - 4}{x - 4}$

Step 5

Divide the expression using synthetic division to determine if it is a factor of the polynomial. Since

x - 4

$x - 4$ divides evenly into

x^{3} - x^{2} - 11 x - 4

$x^{3} - x^{2} - 11 x - 4$ ,

x - 4

$x - 4$ is a factor of the polynomial and there is a remaining polynomial of

x^{2} + 3 x + 1

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

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Step 5.1

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

$4$ $4$	$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$	$- 4$ $- 4$

Step 5.2

The first number in the dividend

(1)

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

$4$ $4$	$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$	$- 4$ $- 4$

	$1$ $1$

Step 5.3

Multiply the newest entry in the result

(1)

$(1)$ by the divisor

(4)

$(4)$ and place the result of

(4)

$(4)$ under the next term in the dividend

(- 1)

$(- 1)$ .

$4$ $4$	$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$	$- 4$ $- 4$
		$4$ $4$
	$1$ $1$

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.

$4$ $4$	$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$	$- 4$ $- 4$
		$4$ $4$
	$1$ $1$	$3$ $3$

Step 5.5

Multiply the newest entry in the result

(3)

$(3)$ by the divisor

(4)

$(4)$ and place the result of

(12)

$(12)$ under the next term in the dividend

(- 11)

$(- 11)$ .

$4$ $4$	$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$	$- 4$ $- 4$
		$4$ $4$	$12$ $12$
	$1$ $1$	$3$ $3$

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.

$4$ $4$	$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$	$- 4$ $- 4$
		$4$ $4$	$12$ $12$
	$1$ $1$	$3$ $3$	$1$ $1$

Step 5.7

Multiply the newest entry in the result

(1)

$(1)$ by the divisor

(4)

$(4)$ and place the result of

(4)

$(4)$ under the next term in the dividend

(- 4)

$(- 4)$ .

$4$ $4$	$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$	$- 4$ $- 4$
		$4$ $4$	$12$ $12$	$4$ $4$
	$1$ $1$	$3$ $3$	$1$ $1$

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.

$4$ $4$	$1$ $1$	$- 1$ $- 1$	$- 11$ $- 11$	$- 4$ $- 4$
		$4$ $4$	$12$ $12$	$4$ $4$
	$1$ $1$	$3$ $3$	$1$ $1$	$0$ $0$

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

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

Step 5.10

Simplify the quotient polynomial.

x^{2} + 3 x + 1

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

x^{2} + 3 x + 1

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

Step 6

Find all the possible roots for

x^{2} + 3 x + 1

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

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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}

$\frac{p}{q}$ where

p

$p$ is a factor of the constant and

q

$q$ is a factor of the leading coefficient.

p = \pm 1

$p = \pm 1$

q = \pm 1

$q = \pm 1$

Step 6.2

Find every combination of

\pm \frac{p}{q}

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

\pm 1

$\pm 1$

\pm 1

$\pm 1$

Step 7

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

x^{2} + 3 x + 1

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

Step 8

The factored polynomial is

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

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

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

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

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