Precalculus Examples

$2 x^{4} + 7 x^{3} - 87 x^{2} - 395 x - 175$

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.

$p = \pm 1, \pm 5, \pm 7, \pm 25, \pm 35, \pm 175$

$q = \pm 1, \pm 2$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm 5, \pm \frac{5}{2}, \pm 7, \pm \frac{7}{2}, \pm 25, \pm \frac{25}{2}, \pm 35, \pm \frac{35}{2}, \pm 175, \pm \frac{175}{2}$

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.

$2 {(- \frac{1}{2})}^{4} + 7 {(- \frac{1}{2})}^{3} - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

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 $- \frac{1}{2}$ .

$2 ({(- 1)}^{4} {(\frac{1}{2})}^{4}) + 7 {(- \frac{1}{2})}^{3} - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.1.2

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

$2 ({(- 1)}^{4} \frac{1^{4}}{2^{4}}) + 7 {(- \frac{1}{2})}^{3} - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.2

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

$2 (1 \frac{1^{4}}{2^{4}}) + 7 {(- \frac{1}{2})}^{3} - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.3

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

$2 \frac{1^{4}}{2^{4}} + 7 {(- \frac{1}{2})}^{3} - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.4

One to any power is one.

$2 \frac{1}{2^{4}} + 7 {(- \frac{1}{2})}^{3} - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.5

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

$2 (\frac{1}{16}) + 7 {(- \frac{1}{2})}^{3} - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.6

Cancel the common factor of $2$ .

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Step 4.1.6.1

Factor $2$ out of $16$ .

$2 \frac{1}{2 (8)} + 7 {(- \frac{1}{2})}^{3} - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.6.2

Cancel the common factor.

$2 \frac{1}{2 \cdot 8} + 7 {(- \frac{1}{2})}^{3} - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.6.3

Rewrite the expression.

$\frac{1}{8} + 7 {(- \frac{1}{2})}^{3} - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

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 $- \frac{1}{2}$ .

$\frac{1}{8} + 7 ({(- 1)}^{3} {(\frac{1}{2})}^{3}) - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.7.2

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

$\frac{1}{8} + 7 ({(- 1)}^{3} \frac{1^{3}}{2^{3}}) - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.8

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

$\frac{1}{8} + 7 (- \frac{1^{3}}{2^{3}}) - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.9

One to any power is one.

$\frac{1}{8} + 7 (- \frac{1}{2^{3}}) - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.10

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

$\frac{1}{8} + 7 (- \frac{1}{8}) - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.11

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

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Step 4.1.11.1

Multiply $- 1$ by $7$ .

$\frac{1}{8} - 7 (\frac{1}{8}) - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.11.2

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

$\frac{1}{8} + \frac{- 7}{8} - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.12

Move the negative in front of the fraction.

$\frac{1}{8} - \frac{7}{8} - 87 {(- \frac{1}{2})}^{2} - 395 (- \frac{1}{2}) - 175$

Step 4.1.13

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

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Step 4.1.13.1

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

$\frac{1}{8} - \frac{7}{8} - 87 ({(- 1)}^{2} {(\frac{1}{2})}^{2}) - 395 (- \frac{1}{2}) - 175$

Step 4.1.13.2

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

$\frac{1}{8} - \frac{7}{8} - 87 ({(- 1)}^{2} \frac{1^{2}}{2^{2}}) - 395 (- \frac{1}{2}) - 175$

Step 4.1.14

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

$\frac{1}{8} - \frac{7}{8} - 87 (1 \frac{1^{2}}{2^{2}}) - 395 (- \frac{1}{2}) - 175$

Step 4.1.15

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

$\frac{1}{8} - \frac{7}{8} - 87 \frac{1^{2}}{2^{2}} - 395 (- \frac{1}{2}) - 175$

Step 4.1.16

One to any power is one.

$\frac{1}{8} - \frac{7}{8} - 87 \frac{1}{2^{2}} - 395 (- \frac{1}{2}) - 175$

Step 4.1.17

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

$\frac{1}{8} - \frac{7}{8} - 87 (\frac{1}{4}) - 395 (- \frac{1}{2}) - 175$

Step 4.1.18

Combine $- 87$ and $\frac{1}{4}$ .

$\frac{1}{8} - \frac{7}{8} + \frac{- 87}{4} - 395 (- \frac{1}{2}) - 175$

Step 4.1.19

Move the negative in front of the fraction.

$\frac{1}{8} - \frac{7}{8} - \frac{87}{4} - 395 (- \frac{1}{2}) - 175$

Step 4.1.20

Multiply $- 395 (- \frac{1}{2})$ .

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Step 4.1.20.1

Multiply $- 1$ by $- 395$ .

$\frac{1}{8} - \frac{7}{8} - \frac{87}{4} + 395 (\frac{1}{2}) - 175$

Step 4.1.20.2

Combine $395$ and $\frac{1}{2}$ .

$\frac{1}{8} - \frac{7}{8} - \frac{87}{4} + \frac{395}{2} - 175$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine the numerators over the common denominator.

$- 175 + \frac{1 - 7}{8} + \frac{- 87}{4} + \frac{395}{2}$

Step 4.2.2

Subtract $7$ from $1$ .

$- 175 + \frac{- 6}{8} + \frac{- 87}{4} + \frac{395}{2}$

Step 4.3

Find the common denominator.

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Step 4.3.1

Write $- 175$ as a fraction with denominator $1$ .

$\frac{- 175}{1} + \frac{- 6}{8} + \frac{- 87}{4} + \frac{395}{2}$

Step 4.3.2

Multiply $\frac{- 175}{1}$ by $\frac{8}{8}$ .

$\frac{- 175}{1} \cdot \frac{8}{8} + \frac{- 6}{8} + \frac{- 87}{4} + \frac{395}{2}$

Step 4.3.3

Multiply $\frac{- 175}{1}$ by $\frac{8}{8}$ .

$\frac{- 175 \cdot 8}{8} + \frac{- 6}{8} + \frac{- 87}{4} + \frac{395}{2}$

Step 4.3.4

Multiply $\frac{- 87}{4}$ by $\frac{2}{2}$ .

$\frac{- 175 \cdot 8}{8} + \frac{- 6}{8} + \frac{- 87}{4} \cdot \frac{2}{2} + \frac{395}{2}$

Step 4.3.5

Multiply $\frac{- 87}{4}$ by $\frac{2}{2}$ .

$\frac{- 175 \cdot 8}{8} + \frac{- 6}{8} + \frac{- 87 \cdot 2}{4 \cdot 2} + \frac{395}{2}$

Step 4.3.6

Multiply $\frac{395}{2}$ by $\frac{4}{4}$ .

$\frac{- 175 \cdot 8}{8} + \frac{- 6}{8} + \frac{- 87 \cdot 2}{4 \cdot 2} + \frac{395}{2} \cdot \frac{4}{4}$

Step 4.3.7

Multiply $\frac{395}{2}$ by $\frac{4}{4}$ .

$\frac{- 175 \cdot 8}{8} + \frac{- 6}{8} + \frac{- 87 \cdot 2}{4 \cdot 2} + \frac{395 \cdot 4}{2 \cdot 4}$

Step 4.3.8

Reorder the factors of $4 \cdot 2$ .

$\frac{- 175 \cdot 8}{8} + \frac{- 6}{8} + \frac{- 87 \cdot 2}{2 \cdot 4} + \frac{395 \cdot 4}{2 \cdot 4}$

Step 4.3.9

Multiply $2$ by $4$ .

$\frac{- 175 \cdot 8}{8} + \frac{- 6}{8} + \frac{- 87 \cdot 2}{8} + \frac{395 \cdot 4}{2 \cdot 4}$

Step 4.3.10

Multiply $2$ by $4$ .

$\frac{- 175 \cdot 8}{8} + \frac{- 6}{8} + \frac{- 87 \cdot 2}{8} + \frac{395 \cdot 4}{8}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{- 175 \cdot 8 - 6 - 87 \cdot 2 + 395 \cdot 4}{8}$

Step 4.5

Simplify each term.

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Step 4.5.1

Multiply $- 175$ by $8$ .

$\frac{- 1400 - 6 - 87 \cdot 2 + 395 \cdot 4}{8}$

Step 4.5.2

Multiply $- 87$ by $2$ .

$\frac{- 1400 - 6 - 174 + 395 \cdot 4}{8}$

Step 4.5.3

Multiply $395$ by $4$ .

$\frac{- 1400 - 6 - 174 + 1580}{8}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Subtract $6$ from $- 1400$ .

$\frac{- 1406 - 174 + 1580}{8}$

Step 4.6.2

Subtract $174$ from $- 1406$ .

$\frac{- 1580 + 1580}{8}$

Step 4.6.3

Add $- 1580$ and $1580$ .

$\frac{0}{8}$

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.

$\frac{2 x^{4} + 7 x^{3} - 87 x^{2} - 395 x - 175}{x + \frac{1}{2}}$

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.

$- \frac{1}{2}$	$2$	$7$	$- 87$	$- 395$	$- 175$

Step 6.2

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

$- \frac{1}{2}$	$2$	$7$	$- 87$	$- 395$	$- 175$

	$2$

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 $(7)$ .

$- \frac{1}{2}$	$2$	$7$	$- 87$	$- 395$	$- 175$
		$- 1$
	$2$

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.

$- \frac{1}{2}$	$2$	$7$	$- 87$	$- 395$	$- 175$
		$- 1$
	$2$	$6$

Step 6.5

Multiply the newest entry in the result $(6)$ by the divisor $(- \frac{1}{2})$ and place the result of $(- 3)$ under the next term in the dividend $(- 87)$ .

$- \frac{1}{2}$	$2$	$7$	$- 87$	$- 395$	$- 175$
		$- 1$	$- 3$
	$2$	$6$

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.

$- \frac{1}{2}$	$2$	$7$	$- 87$	$- 395$	$- 175$
		$- 1$	$- 3$
	$2$	$6$	$- 90$

Step 6.7

Multiply the newest entry in the result $(- 90)$ by the divisor $(- \frac{1}{2})$ and place the result of $(45)$ under the next term in the dividend $(- 395)$ .

$- \frac{1}{2}$	$2$	$7$	$- 87$	$- 395$	$- 175$
		$- 1$	$- 3$	$45$
	$2$	$6$	$- 90$

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.

$- \frac{1}{2}$	$2$	$7$	$- 87$	$- 395$	$- 175$
		$- 1$	$- 3$	$45$
	$2$	$6$	$- 90$	$- 350$

Step 6.9

Multiply the newest entry in the result $(- 350)$ by the divisor $(- \frac{1}{2})$ and place the result of $(175)$ under the next term in the dividend $(- 175)$ .

$- \frac{1}{2}$	$2$	$7$	$- 87$	$- 395$	$- 175$
		$- 1$	$- 3$	$45$	$175$
	$2$	$6$	$- 90$	$- 350$

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.

$- \frac{1}{2}$	$2$	$7$	$- 87$	$- 395$	$- 175$
		$- 1$	$- 3$	$45$	$175$
	$2$	$6$	$- 90$	$- 350$	$0$

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.

$2 x^{3} + 6 x^{2} + (- 90) x - 350$

Step 6.12

Simplify the quotient polynomial.

$2 x^{3} + 6 x^{2} - 90 x - 350$

Step 7

Factor $2$ out of $2 x^{3} + 6 x^{2} - 90 x - 350$ .

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Step 7.1

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

$(x + \frac{1}{2}) (2 (x^{3}) + 6 x^{2} - 90 x - 350)$

Step 7.2

Factor $2$ out of $6 x^{2}$ .

$(x + \frac{1}{2}) (2 (x^{3}) + 2 (3 x^{2}) - 90 x - 350)$

Step 7.3

Factor $2$ out of $- 90 x$ .

$(x + \frac{1}{2}) (2 (x^{3}) + 2 (3 x^{2}) + 2 (- 45 x) - 350)$

Step 7.4

Factor $2$ out of $- 350$ .

$(x + \frac{1}{2}) (2 x^{3} + 2 (3 x^{2}) + 2 (- 45 x) + 2 \cdot - 175)$

Step 7.5

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

$(x + \frac{1}{2}) (2 (x^{3} + 3 x^{2}) + 2 (- 45 x) + 2 \cdot - 175)$

Step 7.6

Factor $2$ out of $2 (x^{3} + 3 x^{2}) + 2 (- 45 x)$ .

$(x + \frac{1}{2}) (2 (x^{3} + 3 x^{2} - 45 x) + 2 \cdot - 175)$

Step 7.7

Factor $2$ out of $2 (x^{3} + 3 x^{2} - 45 x) + 2 \cdot - 175$ .

$(x + \frac{1}{2}) (2 (x^{3} + 3 x^{2} - 45 x - 175))$

Step 8

Factor the left side of the equation.

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Step 8.1

Factor $2 x^{4} + 7 x^{3} - 87 x^{2} - 395 x - 175$ using the rational roots test.

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Step 8.1.1

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.

$p = \pm 1, \pm 175, \pm 5, \pm 35, \pm 7, \pm 25$

$q = \pm 1, \pm 2$

Step 8.1.2

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

$\pm 1, \pm 0.5, \pm 175, \pm 87.5, \pm 5, \pm 2.5, \pm 35, \pm 17.5, \pm 7, \pm 3.5, \pm 25, \pm 12.5$

Step 8.1.3

Substitute $- 0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.5$ is a root of the polynomial.

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Step 8.1.3.1

Substitute $- 0.5$ into the polynomial.

$2 {(- 0.5)}^{4} + 7 {(- 0.5)}^{3} - 87 {(- 0.5)}^{2} - 395 \cdot - 0.5 - 175$

Step 8.1.3.2

Raise $- 0.5$ to the power of $4$ .

$2 \cdot 0.0625 + 7 {(- 0.5)}^{3} - 87 {(- 0.5)}^{2} - 395 \cdot - 0.5 - 175$

Step 8.1.3.3

Multiply $2$ by $0.0625$ .

$0.125 + 7 {(- 0.5)}^{3} - 87 {(- 0.5)}^{2} - 395 \cdot - 0.5 - 175$

Step 8.1.3.4

Raise $- 0.5$ to the power of $3$ .

$0.125 + 7 \cdot - 0.125 - 87 {(- 0.5)}^{2} - 395 \cdot - 0.5 - 175$

Step 8.1.3.5

Multiply $7$ by $- 0.125$ .

$0.125 - 0.875 - 87 {(- 0.5)}^{2} - 395 \cdot - 0.5 - 175$

Step 8.1.3.6

Subtract $0.875$ from $0.125$ .

$- 0.75 - 87 {(- 0.5)}^{2} - 395 \cdot - 0.5 - 175$

Step 8.1.3.7

Raise $- 0.5$ to the power of $2$ .

$- 0.75 - 87 \cdot 0.25 - 395 \cdot - 0.5 - 175$

Step 8.1.3.8

Multiply $- 87$ by $0.25$ .

$- 0.75 - 21.75 - 395 \cdot - 0.5 - 175$

Step 8.1.3.9

Subtract $21.75$ from $- 0.75$ .

$- 22.5 - 395 \cdot - 0.5 - 175$

Step 8.1.3.10

Multiply $- 395$ by $- 0.5$ .

$- 22.5 + 197.5 - 175$

Step 8.1.3.11

Add $- 22.5$ and $197.5$ .

$175 - 175$

Step 8.1.3.12

Subtract $175$ from $175$ .

$0$

Step 8.1.4

Since $- 0.5$ is a known root, divide the polynomial by $2 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{4} + 7 x^{3} - 87 x^{2} - 395 x - 175}{2 x + 1}$

Step 8.1.5

Divide $2 x^{4} + 7 x^{3} - 87 x^{2} - 395 x - 175$ by $2 x + 1$ .

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Step 8.1.5.1

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

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$87 x^{2}$

$395 x$

$175$

Step 8.1.5.2

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$87 x^{2}$

$395 x$

$175$

Step 8.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$87 x^{2}$

$395 x$

$175$

$2 x^{4}$

$x^{3}$

Step 8.1.5.4

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

				$x^{3}$
$2 x$	+	$1$		$2 x^{4}$	+	$7 x^{3}$	-	$87 x^{2}$	-	$395 x$	-	$175$
			-	$2 x^{4}$	-	$x^{3}$

Step 8.1.5.5

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$87 x^{2}$

$395 x$

$175$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

Step 8.1.5.6

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$87 x^{2}$

$395 x$

$175$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$87 x^{2}$

Step 8.1.5.7

Divide the highest order term in the dividend $6 x^{3}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$3 x^{2}$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$87 x^{2}$

$395 x$

$175$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$87 x^{2}$

Step 8.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$87 x^{2}$

$395 x$

$175$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$87 x^{2}$

$6 x^{3}$

$3 x^{2}$

Step 8.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} + 3 x^{2}$

$x^{3}$

$3 x^{2}$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$87 x^{2}$

$395 x$

$175$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$87 x^{2}$

$6 x^{3}$

$3 x^{2}$

Step 8.1.5.10

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

$x^{3}$

$3 x^{2}$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$87 x^{2}$

$395 x$

$175$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$87 x^{2}$

$6 x^{3}$

$3 x^{2}$

$90 x^{2}$

Step 8.1.5.11

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

$x^{3}$

$3 x^{2}$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$87 x^{2}$

$395 x$

$175$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$87 x^{2}$

$6 x^{3}$

$3 x^{2}$

$90 x^{2}$

$395 x$

Step 8.1.5.12

Divide the highest order term in the dividend $- 90 x^{2}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$3 x^{2}$

$45 x$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$87 x^{2}$

$395 x$

$175$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$87 x^{2}$

$6 x^{3}$

$3 x^{2}$

$90 x^{2}$

$395 x$

Step 8.1.5.13

Multiply the new quotient term by the divisor.

				$x^{3}$	+	$3 x^{2}$	-	$45 x$
$2 x$	+	$1$		$2 x^{4}$	+	$7 x^{3}$	-	$87 x^{2}$	-	$395 x$	-	$175$
			-	$2 x^{4}$	-	$x^{3}$
					+	$6 x^{3}$	-	$87 x^{2}$
					-	$6 x^{3}$	-	$3 x^{2}$
							-	$90 x^{2}$	-	$395 x$
							-	$90 x^{2}$	-	$45 x$

Step 8.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 90 x^{2} - 45 x$

$x^{3}$

$3 x^{2}$

$45 x$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$87 x^{2}$

$395 x$

$175$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$87 x^{2}$

$6 x^{3}$

$3 x^{2}$

$90 x^{2}$

$395 x$

$90 x^{2}$

$45 x$

Step 8.1.5.15

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

$x^{3}$

$3 x^{2}$

$45 x$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$87 x^{2}$

$395 x$

$175$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$87 x^{2}$

$6 x^{3}$

$3 x^{2}$

$90 x^{2}$

$395 x$

$90 x^{2}$

$45 x$

$350 x$

Step 8.1.5.16

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

$x^{3}$

$3 x^{2}$

$45 x$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$87 x^{2}$

$395 x$

$175$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$87 x^{2}$

$6 x^{3}$

$3 x^{2}$

$90 x^{2}$

$395 x$

$90 x^{2}$

$45 x$

$350 x$

$175$

Step 8.1.5.17

Divide the highest order term in the dividend $- 350 x$ by the highest order term in divisor $2 x$ .

				$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
$2 x$	+	$1$		$2 x^{4}$	+	$7 x^{3}$	-	$87 x^{2}$	-	$395 x$	-	$175$
			-	$2 x^{4}$	-	$x^{3}$
					+	$6 x^{3}$	-	$87 x^{2}$
					-	$6 x^{3}$	-	$3 x^{2}$
							-	$90 x^{2}$	-	$395 x$
							+	$90 x^{2}$	+	$45 x$
									-	$350 x$	-	$175$

Step 8.1.5.18

Multiply the new quotient term by the divisor.

				$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
$2 x$	+	$1$		$2 x^{4}$	+	$7 x^{3}$	-	$87 x^{2}$	-	$395 x$	-	$175$
			-	$2 x^{4}$	-	$x^{3}$
					+	$6 x^{3}$	-	$87 x^{2}$
					-	$6 x^{3}$	-	$3 x^{2}$
							-	$90 x^{2}$	-	$395 x$
							+	$90 x^{2}$	+	$45 x$
									-	$350 x$	-	$175$
									-	$350 x$	-	$175$

Step 8.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 350 x - 175$

				$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
$2 x$	+	$1$		$2 x^{4}$	+	$7 x^{3}$	-	$87 x^{2}$	-	$395 x$	-	$175$
			-	$2 x^{4}$	-	$x^{3}$
					+	$6 x^{3}$	-	$87 x^{2}$
					-	$6 x^{3}$	-	$3 x^{2}$
							-	$90 x^{2}$	-	$395 x$
							+	$90 x^{2}$	+	$45 x$
									-	$350 x$	-	$175$
									+	$350 x$	+	$175$

Step 8.1.5.20

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

				$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
$2 x$	+	$1$		$2 x^{4}$	+	$7 x^{3}$	-	$87 x^{2}$	-	$395 x$	-	$175$
			-	$2 x^{4}$	-	$x^{3}$
					+	$6 x^{3}$	-	$87 x^{2}$
					-	$6 x^{3}$	-	$3 x^{2}$
							-	$90 x^{2}$	-	$395 x$
							+	$90 x^{2}$	+	$45 x$
									-	$350 x$	-	$175$
									+	$350 x$	+	$175$
												$0$

Step 8.1.5.21

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

$x^{3} + 3 x^{2} - 45 x - 175$

Step 8.1.6

Write $2 x^{4} + 7 x^{3} - 87 x^{2} - 395 x - 175$ as a set of factors.

$(2 x + 1) (x^{3} + 3 x^{2} - 45 x - 175) = 0$

Step 8.2

Factor $x^{3} + 3 x^{2} - 45 x - 175$ using the rational roots test.

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Step 8.2.1

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.

$p = \pm 1, \pm 175, \pm 5, \pm 35, \pm 7, \pm 25$

$q = \pm 1$

Step 8.2.2

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

$\pm 1, \pm 175, \pm 5, \pm 35, \pm 7, \pm 25$

Step 8.2.3

Substitute $- 5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 5$ is a root of the polynomial.

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Step 8.2.3.1

Substitute $- 5$ into the polynomial.

${(- 5)}^{3} + 3 {(- 5)}^{2} - 45 \cdot - 5 - 175$

Step 8.2.3.2

Raise $- 5$ to the power of $3$ .

$- 125 + 3 {(- 5)}^{2} - 45 \cdot - 5 - 175$

Step 8.2.3.3

Raise $- 5$ to the power of $2$ .

$- 125 + 3 \cdot 25 - 45 \cdot - 5 - 175$

Step 8.2.3.4

Multiply $3$ by $25$ .

$- 125 + 75 - 45 \cdot - 5 - 175$

Step 8.2.3.5

Add $- 125$ and $75$ .

$- 50 - 45 \cdot - 5 - 175$

Step 8.2.3.6

Multiply $- 45$ by $- 5$ .

$- 50 + 225 - 175$

Step 8.2.3.7

Add $- 50$ and $225$ .

$175 - 175$

Step 8.2.3.8

Subtract $175$ from $175$ .

$0$

Step 8.2.4

Since $- 5$ is a known root, divide the polynomial by $x + 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 3 x^{2} - 45 x - 175}{x + 5}$

Step 8.2.5

Divide $x^{3} + 3 x^{2} - 45 x - 175$ by $x + 5$ .

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Step 8.2.5.1

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

$x$

$5$

$x^{3}$

$3 x^{2}$

$45 x$

$175$

Step 8.2.5.2

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

					$x^{2}$
	$x$	+	$5$		$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$

Step 8.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
			+	$x^{3}$	+	$5 x^{2}$

Step 8.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 5 x^{2}$

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
			-	$x^{3}$	-	$5 x^{2}$

Step 8.2.5.5

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

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
			-	$x^{3}$	-	$5 x^{2}$
					-	$2 x^{2}$

Step 8.2.5.6

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

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
			-	$x^{3}$	-	$5 x^{2}$
					-	$2 x^{2}$	-	$45 x$

Step 8.2.5.7

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

				$x^{2}$	-	$2 x$
$x$	+	$5$		$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
			-	$x^{3}$	-	$5 x^{2}$
					-	$2 x^{2}$	-	$45 x$

Step 8.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$
$x$	+	$5$		$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
			-	$x^{3}$	-	$5 x^{2}$
					-	$2 x^{2}$	-	$45 x$
					-	$2 x^{2}$	-	$10 x$

Step 8.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} - 10 x$

				$x^{2}$	-	$2 x$
$x$	+	$5$		$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
			-	$x^{3}$	-	$5 x^{2}$
					-	$2 x^{2}$	-	$45 x$
					+	$2 x^{2}$	+	$10 x$

Step 8.2.5.10

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

				$x^{2}$	-	$2 x$
$x$	+	$5$		$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
			-	$x^{3}$	-	$5 x^{2}$
					-	$2 x^{2}$	-	$45 x$
					+	$2 x^{2}$	+	$10 x$
							-	$35 x$

Step 8.2.5.11

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

				$x^{2}$	-	$2 x$
$x$	+	$5$		$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
			-	$x^{3}$	-	$5 x^{2}$
					-	$2 x^{2}$	-	$45 x$
					+	$2 x^{2}$	+	$10 x$
							-	$35 x$	-	$175$

Step 8.2.5.12

Divide the highest order term in the dividend $- 35 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$2 x$	-	$35$
$x$	+	$5$		$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
			-	$x^{3}$	-	$5 x^{2}$
					-	$2 x^{2}$	-	$45 x$
					+	$2 x^{2}$	+	$10 x$
							-	$35 x$	-	$175$

Step 8.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$	-	$35$
$x$	+	$5$		$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
			-	$x^{3}$	-	$5 x^{2}$
					-	$2 x^{2}$	-	$45 x$
					+	$2 x^{2}$	+	$10 x$
							-	$35 x$	-	$175$
							-	$35 x$	-	$175$

Step 8.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 35 x - 175$

				$x^{2}$	-	$2 x$	-	$35$
$x$	+	$5$		$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
			-	$x^{3}$	-	$5 x^{2}$
					-	$2 x^{2}$	-	$45 x$
					+	$2 x^{2}$	+	$10 x$
							-	$35 x$	-	$175$
							+	$35 x$	+	$175$

Step 8.2.5.15

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

				$x^{2}$	-	$2 x$	-	$35$
$x$	+	$5$		$x^{3}$	+	$3 x^{2}$	-	$45 x$	-	$175$
			-	$x^{3}$	-	$5 x^{2}$
					-	$2 x^{2}$	-	$45 x$
					+	$2 x^{2}$	+	$10 x$
							-	$35 x$	-	$175$
							+	$35 x$	+	$175$
										$0$

Step 8.2.5.16

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

$x^{2} - 2 x - 35$

Step 8.2.6

Write $x^{3} + 3 x^{2} - 45 x - 175$ as a set of factors.

$(2 x + 1) ((x + 5) (x^{2} - 2 x - 35)) = 0$

Step 8.3

Factor $x^{2} - 2 x - 35$ using the AC method.

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Step 8.3.1

Factor $x^{2} - 2 x - 35$ using the AC method.

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Step 8.3.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 35$ and whose sum is $- 2$ .

$- 7, 5$

Step 8.3.1.2

Write the factored form using these integers.

$(2 x + 1) ((x + 5) ((x - 7) (x + 5))) = 0$

Step 8.3.2

Remove unnecessary parentheses.

$(2 x + 1) ((x + 5) (x - 7) (x + 5)) = 0$

Step 8.4

Combine exponents.

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Step 8.4.1

Combine exponents.

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Step 8.4.1.1

Raise $x + 5$ to the power of $1$ .

$(2 x + 1) ((x + 5) (x + 5) (x - 7)) = 0$

Step 8.4.1.2

Raise $x + 5$ to the power of $1$ .

$(2 x + 1) ((x + 5) (x + 5) (x - 7)) = 0$

Step 8.4.1.3

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

$(2 x + 1) ({(x + 5)}^{1 + 1} (x - 7)) = 0$

Step 8.4.1.4

Add $1$ and $1$ .

$(2 x + 1) ({(x + 5)}^{2} (x - 7)) = 0$

Step 8.4.2

Remove unnecessary parentheses.

$(2 x + 1) {(x + 5)}^{2} (x - 7) = 0$

Step 9

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

$2 x + 1 = 0$

${(x + 5)}^{2} = 0$

$x - 7 = 0$

Step 10

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

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Step 10.1

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

$2 x + 1 = 0$

Step 10.2

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

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Step 10.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 10.2.2

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

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Step 10.2.2.1

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

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

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of $2$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

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

Step 10.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 10.2.2.3

Simplify the right side.

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Step 10.2.2.3.1

Move the negative in front of the fraction.

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

Step 11

Set ${(x + 5)}^{2}$ equal to $0$ and solve for $x$ .

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Step 11.1

Set ${(x + 5)}^{2}$ equal to $0$ .

${(x + 5)}^{2} = 0$

Step 11.2

Solve ${(x + 5)}^{2} = 0$ for $x$ .

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Step 11.2.1

Set the $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 11.2.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 12

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 12.2

Add $7$ to both sides of the equation.

$x = 7$

Step 13

The final solution is all the values that make $(2 x + 1) {(x + 5)}^{2} (x - 7) = 0$ true.

$x = - \frac{1}{2}, - 5, 7$

Step 14