Precalculus Examples

$x^{5} - 6 x^{4} + 11 x^{3} - 2 x^{2} - 12 x + 8$

Step 1

Regroup terms.

$x^{5} - 2 x^{2} - 12 x - 6 x^{4} + 11 x^{3} + 8$

Step 2

Factor $x$ out of $x^{5} - 2 x^{2} - 12 x$ .

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Step 2.1

Factor $x$ out of $x^{5}$ .

$x \cdot x^{4} - 2 x^{2} - 12 x - 6 x^{4} + 11 x^{3} + 8$

Step 2.2

Factor $x$ out of $- 2 x^{2}$ .

$x \cdot x^{4} + x (- 2 x) - 12 x - 6 x^{4} + 11 x^{3} + 8$

Step 2.3

Factor $x$ out of $- 12 x$ .

$x \cdot x^{4} + x (- 2 x) + x \cdot - 12 - 6 x^{4} + 11 x^{3} + 8$

Step 2.4

Factor $x$ out of $x \cdot x^{4} + x (- 2 x)$ .

$x (x^{4} - 2 x) + x \cdot - 12 - 6 x^{4} + 11 x^{3} + 8$

Step 2.5

Factor $x$ out of $x (x^{4} - 2 x) + x \cdot - 12$ .

$x (x^{4} - 2 x - 12) - 6 x^{4} + 11 x^{3} + 8$

Step 3

Factor.

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Step 3.1

Factor $x^{4} - 2 x - 12$ using the rational roots test.

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Step 3.1.1

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.

$p = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

$q = \pm 1$

Step 3.1.2

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

$\pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

Step 3.1.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 3.1.3.1

Substitute $2$ into the polynomial.

$2^{4} - 2 \cdot 2 - 12$

Step 3.1.3.2

Raise $2$ to the power of $4$ .

$16 - 2 \cdot 2 - 12$

Step 3.1.3.3

Multiply $- 2$ by $2$ .

$16 - 4 - 12$

Step 3.1.3.4

Subtract $4$ from $16$ .

$12 - 12$

Step 3.1.3.5

Subtract $12$ from $12$ .

$0$

Step 3.1.4

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 2 x - 12}{x - 2}$

Step 3.1.5

Divide $x^{4} - 2 x - 12$ by $x - 2$ .

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Step 3.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

Step 3.1.5.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

Step 3.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

Step 3.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - 2 x^{3}$

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

Step 3.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

Step 3.1.5.6

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

Step 3.1.5.7

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$2 x^{2}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

Step 3.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

Step 3.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} - 4 x^{2}$

$x^{3}$

$2 x^{2}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

Step 3.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$2 x^{2}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

Step 3.1.5.11

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$2 x^{2}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x$

Step 3.1.5.12

Divide the highest order term in the dividend $4 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x$

Step 3.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x$

$4 x^{2}$

$8 x$

Step 3.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{2} - 8 x$

$x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x$

$4 x^{2}$

$8 x$

Step 3.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x$

$4 x^{2}$

$8 x$

$6 x$

Step 3.1.5.16

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x$

$4 x^{2}$

$8 x$

$6 x$

$12$

Step 3.1.5.17

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $x$ .

$x^{3}$

$2 x^{2}$

$4 x$

$6$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x$

$4 x^{2}$

$8 x$

$6 x$

$12$

Step 3.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$4 x$

$6$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x$

$4 x^{2}$

$8 x$

$6 x$

$12$

$6 x$

$12$

Step 3.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $6 x - 12$

$x^{3}$

$2 x^{2}$

$4 x$

$6$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x$

$4 x^{2}$

$8 x$

$6 x$

$12$

$6 x$

$12$

Step 3.1.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$2 x^{2}$

$4 x$

$6$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$12$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x$

$4 x^{2}$

$8 x$

$6 x$

$12$

$6 x$

$12$

$0$

Step 3.1.5.21

Since the remander is $0$ , the final answer is the quotient.

$x^{3} + 2 x^{2} + 4 x + 6$

Step 3.1.6

Write $x^{4} - 2 x - 12$ as a set of factors.

$x ((x - 2) (x^{3} + 2 x^{2} + 4 x + 6)) - 6 x^{4} + 11 x^{3} + 8$

Step 3.2

Remove unnecessary parentheses.

$x (x - 2) (x^{3} + 2 x^{2} + 4 x + 6) - 6 x^{4} + 11 x^{3} + 8$

Step 4

Factor $- 6 x^{4} + 11 x^{3} + 8$ using the rational roots test.

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Step 4.1

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.

$p = \pm 1, \pm 8, \pm 2, \pm 4$

$q = \pm 1, \pm 6, \pm 2, \pm 3$

Step 4.2

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

$\pm 1, \pm 0.1\overline{6}, \pm 0.5, \pm 0.\overline{3}, \pm 8, \pm 1.\overline{3}, \pm 4, \pm 2.\overline{6}, \pm 2, \pm 0.\overline{6}$

Step 4.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 4.3.1

Substitute $2$ into the polynomial.

$- 6 \cdot 2^{4} + 11 \cdot 2^{3} + 8$

Step 4.3.2

Raise $2$ to the power of $4$ .

$- 6 \cdot 16 + 11 \cdot 2^{3} + 8$

Step 4.3.3

Multiply $- 6$ by $16$ .

$- 96 + 11 \cdot 2^{3} + 8$

Step 4.3.4

Raise $2$ to the power of $3$ .

$- 96 + 11 \cdot 8 + 8$

Step 4.3.5

Multiply $11$ by $8$ .

$- 96 + 88 + 8$

Step 4.3.6

Add $- 96$ and $88$ .

$- 8 + 8$

Step 4.3.7

Add $- 8$ and $8$ .

$0$

Step 4.4

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- 6 x^{4} + 11 x^{3} + 8}{x - 2}$

Step 4.5

Divide $- 6 x^{4} + 11 x^{3} + 8$ by $x - 2$ .

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Step 4.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

Step 4.5.2

Divide the highest order term in the dividend $- 6 x^{4}$ by the highest order term in divisor $x$ .

$6 x^{3}$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

Step 4.5.3

Multiply the new quotient term by the divisor.

$6 x^{3}$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

Step 4.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{4} + 12 x^{3}$

$6 x^{3}$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

Step 4.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$6 x^{3}$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

Step 4.5.6

Pull the next terms from the original dividend down into the current dividend.

$6 x^{3}$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

Step 4.5.7

Divide the highest order term in the dividend $- x^{3}$ by the highest order term in divisor $x$ .

$6 x^{3}$

$x^{2}$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

Step 4.5.8

Multiply the new quotient term by the divisor.

$6 x^{3}$

$x^{2}$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

Step 4.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{3} + 2 x^{2}$

$6 x^{3}$

$x^{2}$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

Step 4.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$6 x^{3}$

$x^{2}$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

Step 4.5.11

Pull the next terms from the original dividend down into the current dividend.

$6 x^{3}$

$x^{2}$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

Step 4.5.12

Divide the highest order term in the dividend $- 2 x^{2}$ by the highest order term in divisor $x$ .

$6 x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

Step 4.5.13

Multiply the new quotient term by the divisor.

$6 x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

Step 4.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} + 4 x$

$6 x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

Step 4.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$6 x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

Step 4.5.16

Pull the next terms from the original dividend down into the current dividend.

$6 x^{3}$

$x^{2}$

$2 x$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

$8$

Step 4.5.17

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $x$ .

$6 x^{3}$

$x^{2}$

$2 x$

$4$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

$8$

Step 4.5.18

Multiply the new quotient term by the divisor.

$6 x^{3}$

$x^{2}$

$2 x$

$4$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

$8$

$4 x$

$8$

Step 4.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x + 8$

$6 x^{3}$

$x^{2}$

$2 x$

$4$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

$8$

$4 x$

$8$

Step 4.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$6 x^{3}$

$x^{2}$

$2 x$

$4$

$x$

$2$

$6 x^{4}$

$11 x^{3}$

$0 x^{2}$

$0 x$

$8$

$6 x^{4}$

$12 x^{3}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

$8$

$4 x$

$8$

$0$

Step 4.5.21

Since the remander is $0$ , the final answer is the quotient.

$- 6 x^{3} - x^{2} - 2 x - 4$

Step 4.6

Write $- 6 x^{4} + 11 x^{3} + 8$ as a set of factors.

$x (x - 2) (x^{3} + 2 x^{2} + 4 x + 6) + (x - 2) (- 6 x^{3} - x^{2} - 2 x - 4)$

Step 5

Factor $x - 2$ out of $x (x - 2) (x^{3} + 2 x^{2} + 4 x + 6) + (x - 2) (- 6 x^{3} - x^{2} - 2 x - 4)$ .

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Step 5.1

Factor $x - 2$ out of $x (x - 2) (x^{3} + 2 x^{2} + 4 x + 6)$ .

$(x - 2) (x (x^{3} + 2 x^{2} + 4 x + 6)) + (x - 2) (- 6 x^{3} - x^{2} - 2 x - 4)$

Step 5.2

Factor $x - 2$ out of $(x - 2) (x (x^{3} + 2 x^{2} + 4 x + 6)) + (x - 2) (- 6 x^{3} - x^{2} - 2 x - 4)$ .

$(x - 2) (x (x^{3} + 2 x^{2} + 4 x + 6) - 6 x^{3} - x^{2} - 2 x - 4)$

Step 6

Apply the distributive property.

$(x - 2) (x \cdot x^{3} + x (2 x^{2}) + x (4 x) + x \cdot 6 - 6 x^{3} - x^{2} - 2 x - 4)$

Step 7

Simplify.

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Step 7.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 7.1.1

Multiply $x$ by $x^{3}$ .

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Step 7.1.1.1

Raise $x$ to the power of $1$ .

$(x - 2) (x^{1} x^{3} + x (2 x^{2}) + x (4 x) + x \cdot 6 - 6 x^{3} - x^{2} - 2 x - 4)$

Step 7.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x - 2) (x^{1 + 3} + x (2 x^{2}) + x (4 x) + x \cdot 6 - 6 x^{3} - x^{2} - 2 x - 4)$

Step 7.1.2

Add $1$ and $3$ .

$(x - 2) (x^{4} + x (2 x^{2}) + x (4 x) + x \cdot 6 - 6 x^{3} - x^{2} - 2 x - 4)$

Step 7.2

Rewrite using the commutative property of multiplication.

$(x - 2) (x^{4} + 2 x \cdot x^{2} + x (4 x) + x \cdot 6 - 6 x^{3} - x^{2} - 2 x - 4)$

Step 7.3

Rewrite using the commutative property of multiplication.

$(x - 2) (x^{4} + 2 x \cdot x^{2} + 4 x \cdot x + x \cdot 6 - 6 x^{3} - x^{2} - 2 x - 4)$

Step 7.4

Move $6$ to the left of $x$ .

$(x - 2) (x^{4} + 2 x \cdot x^{2} + 4 x \cdot x + 6 \cdot x - 6 x^{3} - x^{2} - 2 x - 4)$

Step 8

Simplify each term.

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Step 8.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 8.1.1

Move $x^{2}$ .

$(x - 2) (x^{4} + 2 (x^{2} x) + 4 x \cdot x + 6 \cdot x - 6 x^{3} - x^{2} - 2 x - 4)$

Step 8.1.2

Multiply $x^{2}$ by $x$ .

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Step 8.1.2.1

Raise $x$ to the power of $1$ .

$(x - 2) (x^{4} + 2 (x^{2} x^{1}) + 4 x \cdot x + 6 \cdot x - 6 x^{3} - x^{2} - 2 x - 4)$

Step 8.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x - 2) (x^{4} + 2 x^{2 + 1} + 4 x \cdot x + 6 \cdot x - 6 x^{3} - x^{2} - 2 x - 4)$

Step 8.1.3

Add $2$ and $1$ .

$(x - 2) (x^{4} + 2 x^{3} + 4 x \cdot x + 6 \cdot x - 6 x^{3} - x^{2} - 2 x - 4)$

Step 8.2

Multiply $x$ by $x$ by adding the exponents.

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Step 8.2.1

Move $x$ .

$(x - 2) (x^{4} + 2 x^{3} + 4 (x \cdot x) + 6 \cdot x - 6 x^{3} - x^{2} - 2 x - 4)$

Step 8.2.2

Multiply $x$ by $x$ .

$(x - 2) (x^{4} + 2 x^{3} + 4 x^{2} + 6 \cdot x - 6 x^{3} - x^{2} - 2 x - 4)$

$(x - 2) (x^{4} + 2 x^{3} + 4 x^{2} + 6 x - 6 x^{3} - x^{2} - 2 x - 4)$

Step 9

Subtract $6 x^{3}$ from $2 x^{3}$ .

$(x - 2) (x^{4} - 4 x^{3} + 4 x^{2} + 6 x - x^{2} - 2 x - 4)$

Step 10

Subtract $x^{2}$ from $4 x^{2}$ .

$(x - 2) (x^{4} - 4 x^{3} + 3 x^{2} + 6 x - 2 x - 4)$

Step 11

Subtract $2 x$ from $6 x$ .

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