Precalculus Examples

$2 x^{4} - 7 x^{3} - 8 x^{2} + 14 x + 8$

Step 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.

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

$q = \pm 1, \pm 2$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm 2, \pm 4, \pm 8$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

$2 {(- \frac{1}{2})}^{4} - 7 {(- \frac{1}{2})}^{3} - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - \frac{1}{2}$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

$2 ({(- 1)}^{4} {(\frac{1}{2})}^{4}) - 7 {(- \frac{1}{2})}^{3} - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.1.2

Apply the product rule to $\frac{1}{2}$ .

$2 ({(- 1)}^{4} \frac{1^{4}}{2^{4}}) - 7 {(- \frac{1}{2})}^{3} - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.2

Raise $- 1$ to the power of $4$ .

$2 (1 \frac{1^{4}}{2^{4}}) - 7 {(- \frac{1}{2})}^{3} - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.3

Multiply $\frac{1^{4}}{2^{4}}$ by $1$ .

$2 \frac{1^{4}}{2^{4}} - 7 {(- \frac{1}{2})}^{3} - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.4

One to any power is one.

$2 \frac{1}{2^{4}} - 7 {(- \frac{1}{2})}^{3} - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.5

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

$2 (\frac{1}{16}) - 7 {(- \frac{1}{2})}^{3} - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.6

Cancel the common factor of $2$ .

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Step 4.1.6.1

Factor $2$ out of $16$ .

$2 \frac{1}{2 (8)} - 7 {(- \frac{1}{2})}^{3} - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.6.2

Cancel the common factor.

$2 \frac{1}{2 \cdot 8} - 7 {(- \frac{1}{2})}^{3} - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.6.3

Rewrite the expression.

$\frac{1}{8} - 7 {(- \frac{1}{2})}^{3} - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.7.1

Apply the product rule to $- \frac{1}{2}$ .

$\frac{1}{8} - 7 ({(- 1)}^{3} {(\frac{1}{2})}^{3}) - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.7.2

Apply the product rule to $\frac{1}{2}$ .

$\frac{1}{8} - 7 ({(- 1)}^{3} \frac{1^{3}}{2^{3}}) - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.8

Raise $- 1$ to the power of $3$ .

$\frac{1}{8} - 7 (- \frac{1^{3}}{2^{3}}) - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.9

One to any power is one.

$\frac{1}{8} - 7 (- \frac{1}{2^{3}}) - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.10

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

$\frac{1}{8} - 7 (- \frac{1}{8}) - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.11

Multiply $- 7 (- \frac{1}{8})$ .

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Step 4.1.11.1

Multiply $- 1$ by $- 7$ .

$\frac{1}{8} + 7 (\frac{1}{8}) - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.11.2

Combine $7$ and $\frac{1}{8}$ .

$\frac{1}{8} + \frac{7}{8} - 8 {(- \frac{1}{2})}^{2} + 14 (- \frac{1}{2}) + 8$

Step 4.1.12

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.12.1

Apply the product rule to $- \frac{1}{2}$ .

$\frac{1}{8} + \frac{7}{8} - 8 ({(- 1)}^{2} {(\frac{1}{2})}^{2}) + 14 (- \frac{1}{2}) + 8$

Step 4.1.12.2

Apply the product rule to $\frac{1}{2}$ .

$\frac{1}{8} + \frac{7}{8} - 8 ({(- 1)}^{2} \frac{1^{2}}{2^{2}}) + 14 (- \frac{1}{2}) + 8$

Step 4.1.13

Raise $- 1$ to the power of $2$ .

$\frac{1}{8} + \frac{7}{8} - 8 (1 \frac{1^{2}}{2^{2}}) + 14 (- \frac{1}{2}) + 8$

Step 4.1.14

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

$\frac{1}{8} + \frac{7}{8} - 8 \frac{1^{2}}{2^{2}} + 14 (- \frac{1}{2}) + 8$

Step 4.1.15

One to any power is one.

$\frac{1}{8} + \frac{7}{8} - 8 \frac{1}{2^{2}} + 14 (- \frac{1}{2}) + 8$

Step 4.1.16

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

$\frac{1}{8} + \frac{7}{8} - 8 (\frac{1}{4}) + 14 (- \frac{1}{2}) + 8$

Step 4.1.17

Cancel the common factor of $4$ .

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Step 4.1.17.1

Factor $4$ out of $- 8$ .

$\frac{1}{8} + \frac{7}{8} + 4 (- 2) \frac{1}{4} + 14 (- \frac{1}{2}) + 8$

Step 4.1.17.2

Cancel the common factor.

$\frac{1}{8} + \frac{7}{8} + 4 \cdot - 2 \frac{1}{4} + 14 (- \frac{1}{2}) + 8$

Step 4.1.17.3

Rewrite the expression.

$\frac{1}{8} + \frac{7}{8} - 2 + 14 (- \frac{1}{2}) + 8$

Step 4.1.18

Cancel the common factor of $2$ .

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Step 4.1.18.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{1}{8} + \frac{7}{8} - 2 + 14 (\frac{- 1}{2}) + 8$

Step 4.1.18.2

Factor $2$ out of $14$ .

$\frac{1}{8} + \frac{7}{8} - 2 + 2 (7) \frac{- 1}{2} + 8$

Step 4.1.18.3

Cancel the common factor.

$\frac{1}{8} + \frac{7}{8} - 2 + 2 \cdot 7 \frac{- 1}{2} + 8$

Step 4.1.18.4

Rewrite the expression.

$\frac{1}{8} + \frac{7}{8} - 2 + 7 \cdot - 1 + 8$

Step 4.1.19

Multiply $7$ by $- 1$ .

$\frac{1}{8} + \frac{7}{8} - 2 - 7 + 8$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine the numerators over the common denominator.

$- 2 - 7 + 8 + \frac{1 + 7}{8}$

Step 4.2.2

Add $1$ and $7$ .

$- 2 - 7 + 8 + \frac{8}{8}$

Step 4.3

Find the common denominator.

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Step 4.3.1

Write $- 2$ as a fraction with denominator $1$ .

$\frac{- 2}{1} - 7 + 8 + \frac{8}{8}$

Step 4.3.2

Multiply $\frac{- 2}{1}$ by $\frac{8}{8}$ .

$\frac{- 2}{1} \cdot \frac{8}{8} - 7 + 8 + \frac{8}{8}$

Step 4.3.3

Multiply $\frac{- 2}{1}$ by $\frac{8}{8}$ .

$\frac{- 2 \cdot 8}{8} - 7 + 8 + \frac{8}{8}$

Step 4.3.4

Write $- 7$ as a fraction with denominator $1$ .

$\frac{- 2 \cdot 8}{8} + \frac{- 7}{1} + 8 + \frac{8}{8}$

Step 4.3.5

Multiply $\frac{- 7}{1}$ by $\frac{8}{8}$ .

$\frac{- 2 \cdot 8}{8} + \frac{- 7}{1} \cdot \frac{8}{8} + 8 + \frac{8}{8}$

Step 4.3.6

Multiply $\frac{- 7}{1}$ by $\frac{8}{8}$ .

$\frac{- 2 \cdot 8}{8} + \frac{- 7 \cdot 8}{8} + 8 + \frac{8}{8}$

Step 4.3.7

Write $8$ as a fraction with denominator $1$ .

$\frac{- 2 \cdot 8}{8} + \frac{- 7 \cdot 8}{8} + \frac{8}{1} + \frac{8}{8}$

Step 4.3.8

Multiply $\frac{8}{1}$ by $\frac{8}{8}$ .

$\frac{- 2 \cdot 8}{8} + \frac{- 7 \cdot 8}{8} + \frac{8}{1} \cdot \frac{8}{8} + \frac{8}{8}$

Step 4.3.9

Multiply $\frac{8}{1}$ by $\frac{8}{8}$ .

$\frac{- 2 \cdot 8}{8} + \frac{- 7 \cdot 8}{8} + \frac{8 \cdot 8}{8} + \frac{8}{8}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{- 2 \cdot 8 - 7 \cdot 8 + 8 \cdot 8 + 8}{8}$

Step 4.5

Simplify each term.

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Step 4.5.1

Multiply $- 2$ by $8$ .

$\frac{- 16 - 7 \cdot 8 + 8 \cdot 8 + 8}{8}$

Step 4.5.2

Multiply $- 7$ by $8$ .

$\frac{- 16 - 56 + 8 \cdot 8 + 8}{8}$

Step 4.5.3

Multiply $8$ by $8$ .

$\frac{- 16 - 56 + 64 + 8}{8}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Subtract $56$ from $- 16$ .

$\frac{- 72 + 64 + 8}{8}$

Step 4.6.2

Add $- 72$ and $64$ .

$\frac{- 8 + 8}{8}$

Step 4.6.3

Add $- 8$ and $8$ .

$\frac{0}{8}$

Step 4.6.4

Divide $0$ by $8$ .

$0$

Step 5

Since $- \frac{1}{2}$ is a known root, divide the polynomial by $x + \frac{1}{2}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{4} - 7 x^{3} - 8 x^{2} + 14 x + 8}{x + \frac{1}{2}}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

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Step 6.1

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

$- \frac{1}{2}$	$2$	$- 7$	$- 8$	$14$	$8$

Step 6.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$- \frac{1}{2}$	$2$	$- 7$	$- 8$	$14$	$8$

	$2$

Step 6.3

Multiply the newest entry in the result $(2)$ by the divisor $(- \frac{1}{2})$ and place the result of $(- 1)$ under the next term in the dividend $(- 7)$ .

$- \frac{1}{2}$	$2$	$- 7$	$- 8$	$14$	$8$
		$- 1$
	$2$

Step 6.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- \frac{1}{2}$	$2$	$- 7$	$- 8$	$14$	$8$
		$- 1$
	$2$	$- 8$

Step 6.5

Multiply the newest entry in the result $(- 8)$ by the divisor $(- \frac{1}{2})$ and place the result of $(4)$ under the next term in the dividend $(- 8)$ .

$- \frac{1}{2}$	$2$	$- 7$	$- 8$	$14$	$8$
		$- 1$	$4$
	$2$	$- 8$

Step 6.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- \frac{1}{2}$	$2$	$- 7$	$- 8$	$14$	$8$
		$- 1$	$4$
	$2$	$- 8$	$- 4$

Step 6.7

Multiply the newest entry in the result $(- 4)$ by the divisor $(- \frac{1}{2})$ and place the result of $(2)$ under the next term in the dividend $(14)$ .

$- \frac{1}{2}$	$2$	$- 7$	$- 8$	$14$	$8$
		$- 1$	$4$	$2$
	$2$	$- 8$	$- 4$

Step 6.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- \frac{1}{2}$	$2$	$- 7$	$- 8$	$14$	$8$
		$- 1$	$4$	$2$
	$2$	$- 8$	$- 4$	$16$

Step 6.9

Multiply the newest entry in the result $(16)$ by the divisor $(- \frac{1}{2})$ and place the result of $(- 8)$ under the next term in the dividend $(8)$ .

$- \frac{1}{2}$	$2$	$- 7$	$- 8$	$14$	$8$
		$- 1$	$4$	$2$	$- 8$
	$2$	$- 8$	$- 4$	$16$

Step 6.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- \frac{1}{2}$	$2$	$- 7$	$- 8$	$14$	$8$
		$- 1$	$4$	$2$	$- 8$
	$2$	$- 8$	$- 4$	$16$	$0$

Step 6.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$2 x^{3} + - 8 x^{2} + (- 4) x + 16$

Step 6.12

Simplify the quotient polynomial.

$2 x^{3} - 8 x^{2} - 4 x + 16$

Step 7

Factor $2$ out of $2 x^{3} - 8 x^{2} - 4 x + 16$ .

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Step 7.1

Factor $2$ out of $2 x^{3}$ .

$(x + \frac{1}{2}) (2 (x^{3}) - 8 x^{2} - 4 x + 16)$

Step 7.2

Factor $2$ out of $- 8 x^{2}$ .

$(x + \frac{1}{2}) (2 (x^{3}) + 2 (- 4 x^{2}) - 4 x + 16)$

Step 7.3

Factor $2$ out of $- 4 x$ .

$(x + \frac{1}{2}) (2 (x^{3}) + 2 (- 4 x^{2}) + 2 (- 2 x) + 16)$

Step 7.4

Factor $2$ out of $16$ .

$(x + \frac{1}{2}) (2 x^{3} + 2 (- 4 x^{2}) + 2 (- 2 x) + 2 \cdot 8)$

Step 7.5

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

$(x + \frac{1}{2}) (2 (x^{3} - 4 x^{2}) + 2 (- 2 x) + 2 \cdot 8)$

Step 7.6

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

$(x + \frac{1}{2}) (2 (x^{3} - 4 x^{2} - 2 x) + 2 \cdot 8)$

Step 7.7

Factor $2$ out of $2 (x^{3} - 4 x^{2} - 2 x) + 2 \cdot 8$ .

$(x + \frac{1}{2}) (2 (x^{3} - 4 x^{2} - 2 x + 8))$

Step 8

Factor out the greatest common factor from each group.

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Step 8.1

Group the first two terms and the last two terms.

$(x + \frac{1}{2}) (2 ((x^{3} - 4 x^{2}) - 2 x + 8))$

Step 8.2

Factor out the greatest common factor (GCF) from each group.

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

Step 9

Factor.

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Step 9.1

Factor the polynomial by factoring out the greatest common factor, $x - 4$ .

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

Step 9.2

Remove unnecessary parentheses.

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

Step 10

Factor the left side of the equation.

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Step 10.1

Regroup terms.

$- 7 x^{3} + 14 x + 2 x^{4} - 8 x^{2} + 8 = 0$

Step 10.2

Factor $- 7 x$ out of $- 7 x^{3} + 14 x$ .

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Step 10.2.1

Factor $- 7 x$ out of $- 7 x^{3}$ .

$- 7 x \cdot x^{2} + 14 x + 2 x^{4} - 8 x^{2} + 8 = 0$

Step 10.2.2

Factor $- 7 x$ out of $14 x$ .

$- 7 x \cdot x^{2} - 7 x \cdot - 2 + 2 x^{4} - 8 x^{2} + 8 = 0$

Step 10.2.3

Factor $- 7 x$ out of $- 7 x (x^{2}) - 7 x (- 2)$ .

$- 7 x (x^{2} - 2) + 2 x^{4} - 8 x^{2} + 8 = 0$

Step 10.3

Factor $2$ out of $2 x^{4} - 8 x^{2} + 8$ .

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Step 10.3.1

Factor $2$ out of $2 x^{4}$ .

$- 7 x (x^{2} - 2) + 2 (x^{4}) - 8 x^{2} + 8 = 0$

Step 10.3.2

Factor $2$ out of $- 8 x^{2}$ .

$- 7 x (x^{2} - 2) + 2 (x^{4}) + 2 (- 4 x^{2}) + 8 = 0$

Step 10.3.3

Factor $2$ out of $8$ .

$- 7 x (x^{2} - 2) + 2 x^{4} + 2 (- 4 x^{2}) + 2 \cdot 4 = 0$

Step 10.3.4

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

$- 7 x (x^{2} - 2) + 2 (x^{4} - 4 x^{2}) + 2 \cdot 4 = 0$

Step 10.3.5

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

$- 7 x (x^{2} - 2) + 2 (x^{4} - 4 x^{2} + 4) = 0$

Step 10.4

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$- 7 x (x^{2} - 2) + 2 ({(x^{2})}^{2} - 4 x^{2} + 4) = 0$

Step 10.5

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- 7 x (x^{2} - 2) + 2 (u^{2} - 4 u + 4) = 0$

Step 10.6

Factor using the perfect square rule.

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Step 10.6.1

Rewrite $4$ as $2^{2}$ .

$- 7 x (x^{2} - 2) + 2 (u^{2} - 4 u + 2^{2}) = 0$

Step 10.6.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 u = 2 \cdot u \cdot 2$

Step 10.6.3

Rewrite the polynomial.

$- 7 x (x^{2} - 2) + 2 (u^{2} - 2 \cdot u \cdot 2 + 2^{2}) = 0$

Step 10.6.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 2$ .

$- 7 x (x^{2} - 2) + 2 {(u - 2)}^{2} = 0$

Step 10.7

Replace all occurrences of $u$ with $x^{2}$ .

$- 7 x (x^{2} - 2) + 2 {(x^{2} - 2)}^{2} = 0$

Step 10.8

Factor $x^{2} - 2$ out of $- 7 x (x^{2} - 2) + 2 {(x^{2} - 2)}^{2}$ .

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Step 10.8.1

Factor $x^{2} - 2$ out of $- 7 x (x^{2} - 2)$ .

$(x^{2} - 2) (- 7 x) + 2 {(x^{2} - 2)}^{2} = 0$

Step 10.8.2

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

$(x^{2} - 2) (- 7 x) + (x^{2} - 2) (2 (x^{2} - 2)) = 0$

Step 10.8.3

Factor $x^{2} - 2$ out of $(x^{2} - 2) (- 7 x) + (x^{2} - 2) (2 (x^{2} - 2))$ .

$(x^{2} - 2) (- 7 x + 2 (x^{2} - 2)) = 0$

Step 10.9

Apply the distributive property.

$(x^{2} - 2) (- 7 x + 2 x^{2} + 2 \cdot - 2) = 0$

Step 10.10

Multiply $2$ by $- 2$ .

$(x^{2} - 2) (- 7 x + 2 x^{2} - 4) = 0$

Step 10.11

Reorder terms.

$(x^{2} - 2) (2 x^{2} - 7 x - 4) = 0$

Step 10.12

Factor.

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Step 10.12.1

Factor by grouping.

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Step 10.12.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 4 = - 8$ and whose sum is $b = - 7$ .

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Step 10.12.1.1.1

Factor $- 7$ out of $- 7 x$ .

$(x^{2} - 2) (2 x^{2} - 7 x - 4) = 0$

Step 10.12.1.1.2

Rewrite $- 7$ as $1$ plus $- 8$

$(x^{2} - 2) (2 x^{2} + (1 - 8) x - 4) = 0$

Step 10.12.1.1.3

Apply the distributive property.

$(x^{2} - 2) (2 x^{2} + 1 x - 8 x - 4) = 0$

Step 10.12.1.1.4

Multiply $x$ by $1$ .

$(x^{2} - 2) (2 x^{2} + x - 8 x - 4) = 0$

Step 10.12.1.2

Factor out the greatest common factor from each group.

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Step 10.12.1.2.1

Group the first two terms and the last two terms.

$(x^{2} - 2) ((2 x^{2} + x) - 8 x - 4) = 0$

Step 10.12.1.2.2

Factor out the greatest common factor (GCF) from each group.

$(x^{2} - 2) (x (2 x + 1) - 4 (2 x + 1)) = 0$

Step 10.12.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$(x^{2} - 2) ((2 x + 1) (x - 4)) = 0$

Step 10.12.2

Remove unnecessary parentheses.

$(x^{2} - 2) (2 x + 1) (x - 4) = 0$

Step 11

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} - 2 = 0$

$2 x + 1 = 0$

$x - 4 = 0$

Step 12

Set $x^{2} - 2$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x^{2} - 2$ equal to $0$ .

$x^{2} - 2 = 0$

Step 12.2

Solve $x^{2} - 2 = 0$ for $x$ .

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Step 12.2.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 12.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{2}$

Step 12.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 12.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2}$

Step 12.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2}$

Step 12.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{2}, - \sqrt{2}$

Step 13

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 13.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 13.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 13.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 13.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 13.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 13.2.2.2

Simplify the left side.

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Step 13.2.2.2.1

Cancel the common factor of $2$ .

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Step 13.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 13.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 13.2.2.3

Simplify the right side.

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Step 13.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 14

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 14.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 14.2

Add $4$ to both sides of the equation.

$x = 4$

Step 15

The final solution is all the values that make $(x^{2} - 2) (2 x + 1) (x - 4) = 0$ true.

$x = \sqrt{2}, - \sqrt{2}, - \frac{1}{2}, 4$

Step 16

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{2}, - \sqrt{2}, - \frac{1}{2}, 4$

Decimal Form:

$x = 1.41421356 \dots, - 1.41421356 \dots, - 0.5, 4$

Step 17