Algebra Examples

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

$x - 1$

Step 1

Divide

$\frac{x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4}{x - 1}$ using synthetic division and check if the remainder is equal to

$0$ . If the remainder is equal to

$0$ , it means that

$x - 1$ 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

$x - 1$ 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.

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

Step 1.2

The first number in the dividend

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

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

	$1$

Step 1.3

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

$(- 2)$ .

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

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.

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

Step 1.5

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

$(- 10)$ .

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

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.

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

Step 1.7

Multiply the newest entry in the result

$(- 11)$ by the divisor

$(1)$ and place the result of

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

$(7)$ .

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

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.

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

Step 1.9

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

$(4)$ .

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

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.

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

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.

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

Step 1.12

Simplify the quotient polynomial.

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

Step 2

The remainder from dividing

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

$0$ , which means that

$x - 1$ is a factor for

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

$x - 1$ is a factor for

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

Step 3

Find all the possible roots for

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

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Step 3.1

If a polynomial function has integer coefficients, then every rational zero will have the form

$\frac{p}{q}$ 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

$x - 4$ is a factor of the polynomial

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

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

Step 5

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

$x - 4$ divides evenly into

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

$x - 4$ 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.

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

Step 5.2

The first number in the dividend

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

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

	$1$

Step 5.3

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

$(- 1)$ .

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

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.

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

Step 5.5

Multiply the newest entry in the result

$(3)$ by the divisor

$(4)$ and place the result of

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

$(- 11)$ .

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

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.

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

Step 5.7

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

$(- 4)$ .

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

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.

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

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

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

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

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