Precalculus Examples

$p (x) = 4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9$

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.

$p = \pm 1, \pm 3, \pm 9$

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

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm 9, \pm \frac{9}{2}, \pm \frac{9}{4}$

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.

$4 {(- \frac{3}{2})}^{4} + 8 {(- \frac{3}{2})}^{3} - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - \frac{3}{2}$ 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 $- \frac{3}{2}$ .

$4 ({(- 1)}^{4} {(\frac{3}{2})}^{4}) + 8 {(- \frac{3}{2})}^{3} - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.1.2

Apply the product rule to $\frac{3}{2}$ .

$4 ({(- 1)}^{4} \frac{3^{4}}{2^{4}}) + 8 {(- \frac{3}{2})}^{3} - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.2

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

$4 (1 \frac{3^{4}}{2^{4}}) + 8 {(- \frac{3}{2})}^{3} - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.3

Multiply $\frac{3^{4}}{2^{4}}$ by $1$ .

$4 \frac{3^{4}}{2^{4}} + 8 {(- \frac{3}{2})}^{3} - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.4

Raise $3$ to the power of $4$ .

$4 \frac{81}{2^{4}} + 8 {(- \frac{3}{2})}^{3} - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.5

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

$4 (\frac{81}{16}) + 8 {(- \frac{3}{2})}^{3} - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.6

Cancel the common factor of $4$ .

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Step 4.1.6.1

Factor $4$ out of $16$ .

$4 \frac{81}{4 (4)} + 8 {(- \frac{3}{2})}^{3} - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.6.2

Cancel the common factor.

$4 \frac{81}{4 \cdot 4} + 8 {(- \frac{3}{2})}^{3} - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.6.3

Rewrite the expression.

$\frac{81}{4} + 8 {(- \frac{3}{2})}^{3} - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

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 $- \frac{3}{2}$ .

$\frac{81}{4} + 8 ({(- 1)}^{3} {(\frac{3}{2})}^{3}) - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.7.2

Apply the product rule to $\frac{3}{2}$ .

$\frac{81}{4} + 8 ({(- 1)}^{3} \frac{3^{3}}{2^{3}}) - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.8

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

$\frac{81}{4} + 8 (- \frac{3^{3}}{2^{3}}) - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.9

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

$\frac{81}{4} + 8 (- \frac{27}{2^{3}}) - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.10

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

$\frac{81}{4} + 8 (- \frac{27}{8}) - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.11

Cancel the common factor of $8$ .

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Step 4.1.11.1

Move the leading negative in $- \frac{27}{8}$ into the numerator.

$\frac{81}{4} + 8 (\frac{- 27}{8}) - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.11.2

Cancel the common factor.

$\frac{81}{4} + 8 (\frac{- 27}{8}) - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

Step 4.1.11.3

Rewrite the expression.

$\frac{81}{4} - 27 - 7 {(- \frac{3}{2})}^{2} - 21 (- \frac{3}{2}) - 9$

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 $- \frac{3}{2}$ .

$\frac{81}{4} - 27 - 7 ({(- 1)}^{2} {(\frac{3}{2})}^{2}) - 21 (- \frac{3}{2}) - 9$

Step 4.1.12.2

Apply the product rule to $\frac{3}{2}$ .

$\frac{81}{4} - 27 - 7 ({(- 1)}^{2} \frac{3^{2}}{2^{2}}) - 21 (- \frac{3}{2}) - 9$

Step 4.1.13

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

$\frac{81}{4} - 27 - 7 (1 \frac{3^{2}}{2^{2}}) - 21 (- \frac{3}{2}) - 9$

Step 4.1.14

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$\frac{81}{4} - 27 - 7 \frac{3^{2}}{2^{2}} - 21 (- \frac{3}{2}) - 9$

Step 4.1.15

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

$\frac{81}{4} - 27 - 7 \frac{9}{2^{2}} - 21 (- \frac{3}{2}) - 9$

Step 4.1.16

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

$\frac{81}{4} - 27 - 7 (\frac{9}{4}) - 21 (- \frac{3}{2}) - 9$

Step 4.1.17

Multiply $- 7 (\frac{9}{4})$ .

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Step 4.1.17.1

Combine $- 7$ and $\frac{9}{4}$ .

$\frac{81}{4} - 27 + \frac{- 7 \cdot 9}{4} - 21 (- \frac{3}{2}) - 9$

Step 4.1.17.2

Multiply $- 7$ by $9$ .

$\frac{81}{4} - 27 + \frac{- 63}{4} - 21 (- \frac{3}{2}) - 9$

Step 4.1.18

Move the negative in front of the fraction.

$\frac{81}{4} - 27 - \frac{63}{4} - 21 (- \frac{3}{2}) - 9$

Step 4.1.19

Multiply $- 21 (- \frac{3}{2})$ .

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Step 4.1.19.1

Multiply $- 1$ by $- 21$ .

$\frac{81}{4} - 27 - \frac{63}{4} + 21 (\frac{3}{2}) - 9$

Step 4.1.19.2

Combine $21$ and $\frac{3}{2}$ .

$\frac{81}{4} - 27 - \frac{63}{4} + \frac{21 \cdot 3}{2} - 9$

Step 4.1.19.3

Multiply $21$ by $3$ .

$\frac{81}{4} - 27 - \frac{63}{4} + \frac{63}{2} - 9$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine the numerators over the common denominator.

$- 27 - 9 + \frac{81 - 63}{4} + \frac{63}{2}$

Step 4.2.2

Subtract $63$ from $81$ .

$- 27 - 9 + \frac{18}{4} + \frac{63}{2}$

Step 4.3

Find the common denominator.

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Step 4.3.1

Write $- 27$ as a fraction with denominator $1$ .

$\frac{- 27}{1} - 9 + \frac{18}{4} + \frac{63}{2}$

Step 4.3.2

Multiply $\frac{- 27}{1}$ by $\frac{4}{4}$ .

$\frac{- 27}{1} \cdot \frac{4}{4} - 9 + \frac{18}{4} + \frac{63}{2}$

Step 4.3.3

Multiply $\frac{- 27}{1}$ by $\frac{4}{4}$ .

$\frac{- 27 \cdot 4}{4} - 9 + \frac{18}{4} + \frac{63}{2}$

Step 4.3.4

Write $- 9$ as a fraction with denominator $1$ .

$\frac{- 27 \cdot 4}{4} + \frac{- 9}{1} + \frac{18}{4} + \frac{63}{2}$

Step 4.3.5

Multiply $\frac{- 9}{1}$ by $\frac{4}{4}$ .

$\frac{- 27 \cdot 4}{4} + \frac{- 9}{1} \cdot \frac{4}{4} + \frac{18}{4} + \frac{63}{2}$

Step 4.3.6

Multiply $\frac{- 9}{1}$ by $\frac{4}{4}$ .

$\frac{- 27 \cdot 4}{4} + \frac{- 9 \cdot 4}{4} + \frac{18}{4} + \frac{63}{2}$

Step 4.3.7

Multiply $\frac{63}{2}$ by $\frac{2}{2}$ .

$\frac{- 27 \cdot 4}{4} + \frac{- 9 \cdot 4}{4} + \frac{18}{4} + \frac{63}{2} \cdot \frac{2}{2}$

Step 4.3.8

Multiply $\frac{63}{2}$ by $\frac{2}{2}$ .

$\frac{- 27 \cdot 4}{4} + \frac{- 9 \cdot 4}{4} + \frac{18}{4} + \frac{63 \cdot 2}{2 \cdot 2}$

Step 4.3.9

Multiply $2$ by $2$ .

$\frac{- 27 \cdot 4}{4} + \frac{- 9 \cdot 4}{4} + \frac{18}{4} + \frac{63 \cdot 2}{4}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{- 27 \cdot 4 - 9 \cdot 4 + 18 + 63 \cdot 2}{4}$

Step 4.5

Simplify each term.

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Step 4.5.1

Multiply $- 27$ by $4$ .

$\frac{- 108 - 9 \cdot 4 + 18 + 63 \cdot 2}{4}$

Step 4.5.2

Multiply $- 9$ by $4$ .

$\frac{- 108 - 36 + 18 + 63 \cdot 2}{4}$

Step 4.5.3

Multiply $63$ by $2$ .

$\frac{- 108 - 36 + 18 + 126}{4}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Subtract $36$ from $- 108$ .

$\frac{- 144 + 18 + 126}{4}$

Step 4.6.2

Add $- 144$ and $18$ .

$\frac{- 126 + 126}{4}$

Step 4.6.3

Add $- 126$ and $126$ .

$\frac{0}{4}$

Step 4.6.4

Divide $0$ by $4$ .

$0$

Step 5

Since $- \frac{3}{2}$ is a known root, divide the polynomial by $x + \frac{3}{2}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9}{x + \frac{3}{2}}$

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.

$- \frac{3}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$

Step 6.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- \frac{3}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$

	$4$

Step 6.3

Multiply the newest entry in the result $(4)$ by the divisor $(- \frac{3}{2})$ and place the result of $(- 6)$ under the next term in the dividend $(8)$ .

$- \frac{3}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 6$
	$4$

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.

$- \frac{3}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 6$
	$4$	$2$

Step 6.5

Multiply the newest entry in the result $(2)$ by the divisor $(- \frac{3}{2})$ and place the result of $(- 3)$ under the next term in the dividend $(- 7)$ .

$- \frac{3}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 6$	$- 3$
	$4$	$2$

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.

$- \frac{3}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 6$	$- 3$
	$4$	$2$	$- 10$

Step 6.7

Multiply the newest entry in the result $(- 10)$ by the divisor $(- \frac{3}{2})$ and place the result of $(15)$ under the next term in the dividend $(- 21)$ .

$- \frac{3}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 6$	$- 3$	$15$
	$4$	$2$	$- 10$

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.

$- \frac{3}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 6$	$- 3$	$15$
	$4$	$2$	$- 10$	$- 6$

Step 6.9

Multiply the newest entry in the result $(- 6)$ by the divisor $(- \frac{3}{2})$ and place the result of $(9)$ under the next term in the dividend $(- 9)$ .

$- \frac{3}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 6$	$- 3$	$15$	$9$
	$4$	$2$	$- 10$	$- 6$

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.

$- \frac{3}{2}$	$4$	$8$	$- 7$	$- 21$	$- 9$
		$- 6$	$- 3$	$15$	$9$
	$4$	$2$	$- 10$	$- 6$	$0$

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.

$4 x^{3} + 2 x^{2} + (- 10) x - 6$

Step 6.12

Simplify the quotient polynomial.

$4 x^{3} + 2 x^{2} - 10 x - 6$

Step 7

Factor $2$ out of $4 x^{3} + 2 x^{2} - 10 x - 6$ .

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Step 7.1

Factor $2$ out of $4 x^{3}$ .

$(x + \frac{3}{2}) (2 (2 x^{3}) + 2 x^{2} - 10 x - 6)$

Step 7.2

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

$(x + \frac{3}{2}) (2 (2 x^{3}) + 2 (x^{2}) - 10 x - 6)$

Step 7.3

Factor $2$ out of $- 10 x$ .

$(x + \frac{3}{2}) (2 (2 x^{3}) + 2 (x^{2}) + 2 (- 5 x) - 6)$

Step 7.4

Factor $2$ out of $- 6$ .

$(x + \frac{3}{2}) (2 (2 x^{3}) + 2 x^{2} + 2 (- 5 x) + 2 \cdot - 3)$

Step 7.5

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

$(x + \frac{3}{2}) (2 (2 x^{3} + x^{2}) + 2 (- 5 x) + 2 \cdot - 3)$

Step 7.6

Factor $2$ out of $2 (2 x^{3} + x^{2}) + 2 (- 5 x)$ .

$(x + \frac{3}{2}) (2 (2 x^{3} + x^{2} - 5 x) + 2 \cdot - 3)$

Step 7.7

Factor $2$ out of $2 (2 x^{3} + x^{2} - 5 x) + 2 \cdot - 3$ .

$(x + \frac{3}{2}) (2 (2 x^{3} + x^{2} - 5 x - 3))$

Step 8

Factor the left side of the equation.

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Step 8.1

Factor $4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9$ 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.

$p = \pm 1, \pm 9, \pm 3$

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

Step 8.1.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 9, \pm 2.25, \pm 4.5, \pm 3, \pm 0.75, \pm 1.5$

Step 8.1.3

Substitute $- 1.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1.5$ is a root of the polynomial.

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Step 8.1.3.1

Substitute $- 1.5$ into the polynomial.

$4 {(- 1.5)}^{4} + 8 {(- 1.5)}^{3} - 7 {(- 1.5)}^{2} - 21 \cdot - 1.5 - 9$

Step 8.1.3.2

Raise $- 1.5$ to the power of $4$ .

$4 \cdot 5.0625 + 8 {(- 1.5)}^{3} - 7 {(- 1.5)}^{2} - 21 \cdot - 1.5 - 9$

Step 8.1.3.3

Multiply $4$ by $5.0625$ .

$20.25 + 8 {(- 1.5)}^{3} - 7 {(- 1.5)}^{2} - 21 \cdot - 1.5 - 9$

Step 8.1.3.4

Raise $- 1.5$ to the power of $3$ .

$20.25 + 8 \cdot - 3.375 - 7 {(- 1.5)}^{2} - 21 \cdot - 1.5 - 9$

Step 8.1.3.5

Multiply $8$ by $- 3.375$ .

$20.25 - 27 - 7 {(- 1.5)}^{2} - 21 \cdot - 1.5 - 9$

Step 8.1.3.6

Subtract $27$ from $20.25$ .

$- 6.75 - 7 {(- 1.5)}^{2} - 21 \cdot - 1.5 - 9$

Step 8.1.3.7

Raise $- 1.5$ to the power of $2$ .

$- 6.75 - 7 \cdot 2.25 - 21 \cdot - 1.5 - 9$

Step 8.1.3.8

Multiply $- 7$ by $2.25$ .

$- 6.75 - 15.75 - 21 \cdot - 1.5 - 9$

Step 8.1.3.9

Subtract $15.75$ from $- 6.75$ .

$- 22.5 - 21 \cdot - 1.5 - 9$

Step 8.1.3.10

Multiply $- 21$ by $- 1.5$ .

$- 22.5 + 31.5 - 9$

Step 8.1.3.11

Add $- 22.5$ and $31.5$ .

$9 - 9$

Step 8.1.3.12

Subtract $9$ from $9$ .

$0$

Step 8.1.4

Since $- 1.5$ is a known root, divide the polynomial by $2 x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9}{2 x + 3}$

Step 8.1.5

Divide $4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9$ by $2 x + 3$ .

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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 $0$ .

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

Step 8.1.5.2

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

$2 x^{3}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

Step 8.1.5.3

Multiply the new quotient term by the divisor.

$2 x^{3}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

Step 8.1.5.4

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

				$2 x^{3}$
$2 x$	+	$3$		$4 x^{4}$	+	$8 x^{3}$	-	$7 x^{2}$	-	$21 x$	-	$9$
			-	$4 x^{4}$	-	$6 x^{3}$

Step 8.1.5.5

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

$2 x^{3}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

Step 8.1.5.6

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

$2 x^{3}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

Step 8.1.5.7

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

$2 x^{3}$

$x^{2}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

Step 8.1.5.8

Multiply the new quotient term by the divisor.

$2 x^{3}$

$x^{2}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

Step 8.1.5.9

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

$2 x^{3}$

$x^{2}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

Step 8.1.5.10

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

$2 x^{3}$

$x^{2}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

$10 x^{2}$

Step 8.1.5.11

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

$2 x^{3}$

$x^{2}$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

$10 x^{2}$

$21 x$

Step 8.1.5.12

Divide the highest order term in the dividend $- 10 x^{2}$ by the highest order term in divisor $2 x$ .

$2 x^{3}$

$x^{2}$

$5 x$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

$10 x^{2}$

$21 x$

Step 8.1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{3}$	+	$x^{2}$	-	$5 x$
$2 x$	+	$3$		$4 x^{4}$	+	$8 x^{3}$	-	$7 x^{2}$	-	$21 x$	-	$9$
			-	$4 x^{4}$	-	$6 x^{3}$
					+	$2 x^{3}$	-	$7 x^{2}$
					-	$2 x^{3}$	-	$3 x^{2}$
							-	$10 x^{2}$	-	$21 x$
							-	$10 x^{2}$	-	$15 x$

Step 8.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} - 15 x$

$2 x^{3}$

$x^{2}$

$5 x$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

$10 x^{2}$

$21 x$

$10 x^{2}$

$15 x$

Step 8.1.5.15

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

$2 x^{3}$

$x^{2}$

$5 x$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

$10 x^{2}$

$21 x$

$10 x^{2}$

$15 x$

$6 x$

Step 8.1.5.16

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

$2 x^{3}$

$x^{2}$

$5 x$

$2 x$

$3$

$4 x^{4}$

$8 x^{3}$

$7 x^{2}$

$21 x$

$9$

$4 x^{4}$

$6 x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$3 x^{2}$

$10 x^{2}$

$21 x$

$10 x^{2}$

$15 x$

$6 x$

$9$

Step 8.1.5.17

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $2 x$ .

				$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
$2 x$	+	$3$		$4 x^{4}$	+	$8 x^{3}$	-	$7 x^{2}$	-	$21 x$	-	$9$
			-	$4 x^{4}$	-	$6 x^{3}$
					+	$2 x^{3}$	-	$7 x^{2}$
					-	$2 x^{3}$	-	$3 x^{2}$
							-	$10 x^{2}$	-	$21 x$
							+	$10 x^{2}$	+	$15 x$
									-	$6 x$	-	$9$

Step 8.1.5.18

Multiply the new quotient term by the divisor.

				$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
$2 x$	+	$3$		$4 x^{4}$	+	$8 x^{3}$	-	$7 x^{2}$	-	$21 x$	-	$9$
			-	$4 x^{4}$	-	$6 x^{3}$
					+	$2 x^{3}$	-	$7 x^{2}$
					-	$2 x^{3}$	-	$3 x^{2}$
							-	$10 x^{2}$	-	$21 x$
							+	$10 x^{2}$	+	$15 x$
									-	$6 x$	-	$9$
									-	$6 x$	-	$9$

Step 8.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x - 9$

				$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
$2 x$	+	$3$		$4 x^{4}$	+	$8 x^{3}$	-	$7 x^{2}$	-	$21 x$	-	$9$
			-	$4 x^{4}$	-	$6 x^{3}$
					+	$2 x^{3}$	-	$7 x^{2}$
					-	$2 x^{3}$	-	$3 x^{2}$
							-	$10 x^{2}$	-	$21 x$
							+	$10 x^{2}$	+	$15 x$
									-	$6 x$	-	$9$
									+	$6 x$	+	$9$

Step 8.1.5.20

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

				$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
$2 x$	+	$3$		$4 x^{4}$	+	$8 x^{3}$	-	$7 x^{2}$	-	$21 x$	-	$9$
			-	$4 x^{4}$	-	$6 x^{3}$
					+	$2 x^{3}$	-	$7 x^{2}$
					-	$2 x^{3}$	-	$3 x^{2}$
							-	$10 x^{2}$	-	$21 x$
							+	$10 x^{2}$	+	$15 x$
									-	$6 x$	-	$9$
									+	$6 x$	+	$9$
												$0$

Step 8.1.5.21

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

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

Step 8.1.6

Write $4 x^{4} + 8 x^{3} - 7 x^{2} - 21 x - 9$ as a set of factors.

$(2 x + 3) (2 x^{3} + x^{2} - 5 x - 3) = 0$

Step 8.2

Factor $2 x^{3} + x^{2} - 5 x - 3$ using the rational roots test.

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Step 8.2.1

Factor $2 x^{3} + x^{2} - 5 x - 3$ using the rational roots test.

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Step 8.2.1.1

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.

$p = \pm 1, \pm 3$

$q = \pm 1, \pm 2$

Step 8.2.1.2

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

$\pm 1, \pm 0.5, \pm 3, \pm 1.5$

Step 8.2.1.3

Substitute $- 1.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1.5$ is a root of the polynomial.

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Step 8.2.1.3.1

Substitute $- 1.5$ into the polynomial.

$2 {(- 1.5)}^{3} + {(- 1.5)}^{2} - 5 \cdot - 1.5 - 3$

Step 8.2.1.3.2

Raise $- 1.5$ to the power of $3$ .

$2 \cdot - 3.375 + {(- 1.5)}^{2} - 5 \cdot - 1.5 - 3$

Step 8.2.1.3.3

Multiply $2$ by $- 3.375$ .

$- 6.75 + {(- 1.5)}^{2} - 5 \cdot - 1.5 - 3$

Step 8.2.1.3.4

Raise $- 1.5$ to the power of $2$ .

$- 6.75 + 2.25 - 5 \cdot - 1.5 - 3$

Step 8.2.1.3.5

Add $- 6.75$ and $2.25$ .

$- 4.5 - 5 \cdot - 1.5 - 3$

Step 8.2.1.3.6

Multiply $- 5$ by $- 1.5$ .

$- 4.5 + 7.5 - 3$

Step 8.2.1.3.7

Add $- 4.5$ and $7.5$ .

$3 - 3$

Step 8.2.1.3.8

Subtract $3$ from $3$ .

$0$

Step 8.2.1.4

Since $- 1.5$ is a known root, divide the polynomial by $2 x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

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

Step 8.2.1.5

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

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Step 8.2.1.5.1

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

$2 x$

$3$

$2 x^{3}$

$x^{2}$

$5 x$

$3$

Step 8.2.1.5.2

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

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

Step 8.2.1.5.3

Multiply the new quotient term by the divisor.

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

Step 8.2.1.5.4

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

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

Step 8.2.1.5.5

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

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

Step 8.2.1.5.6

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

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

Step 8.2.1.5.7

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

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

Step 8.2.1.5.8

Multiply the new quotient term by the divisor.

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

Step 8.2.1.5.9

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

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

Step 8.2.1.5.10

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

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

Step 8.2.1.5.11

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

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

Step 8.2.1.5.12

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

				$x^{2}$	-	$x$	-	$1$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$3 x$
							-	$2 x$	-	$3$

Step 8.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	-	$1$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$3 x$
							-	$2 x$	-	$3$
							-	$2 x$	-	$3$

Step 8.2.1.5.14

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

				$x^{2}$	-	$x$	-	$1$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$3 x$
							-	$2 x$	-	$3$
							+	$2 x$	+	$3$

Step 8.2.1.5.15

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

				$x^{2}$	-	$x$	-	$1$
$2 x$	+	$3$		$2 x^{3}$	+	$x^{2}$	-	$5 x$	-	$3$
			-	$2 x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$3 x$
							-	$2 x$	-	$3$
							+	$2 x$	+	$3$
										$0$

Step 8.2.1.5.16

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

$x^{2} - x - 1$

Step 8.2.1.6

Write $2 x^{3} + x^{2} - 5 x - 3$ as a set of factors.

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

Step 8.2.2

Remove unnecessary parentheses.

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

Step 8.3

Combine like factors.

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Step 8.3.1

Raise $2 x + 3$ to the power of $1$ .

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

Step 8.3.2

Raise $2 x + 3$ to the power of $1$ .

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

Step 8.3.3

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

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

Step 8.3.4

Add $1$ and $1$ .

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

Step 9

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

${(2 x + 3)}^{2} = 0$

$x^{2} - x - 1 = 0$

Step 10

Set ${(2 x + 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 10.1

Set ${(2 x + 3)}^{2}$ equal to $0$ .

${(2 x + 3)}^{2} = 0$

Step 10.2

Solve ${(2 x + 3)}^{2} = 0$ for $x$ .

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Step 10.2.1

Set the $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Step 10.2.2

Solve for $x$ .

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Step 10.2.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 10.2.2.2

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 10.2.2.2.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 10.2.2.2.2

Simplify the left side.

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Step 10.2.2.2.2.1

Cancel the common factor of $2$ .

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Step 10.2.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 10.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 10.2.2.2.3

Simplify the right side.

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Step 10.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 11

Set $x^{2} - x - 1$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $x^{2} - x - 1$ equal to $0$ .

$x^{2} - x - 1 = 0$

Step 11.2

Solve $x^{2} - x - 1 = 0$ for $x$ .

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Step 11.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 11.2.2

Substitute the values $a = 1$ , $b = - 1$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 11.2.3

Simplify.

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Step 11.2.3.1

Simplify the numerator.

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Step 11.2.3.1.1

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

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

Step 11.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 11.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 11.2.3.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{1 \pm \sqrt{1 + 4}}{2 \cdot 1}$

Step 11.2.3.1.3

Add $1$ and $4$ .

$x = \frac{1 \pm \sqrt{5}}{2 \cdot 1}$

Step 11.2.3.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm \sqrt{5}}{2}$

Step 11.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 11.2.4.1

Simplify the numerator.

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Step 11.2.4.1.1

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

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

Step 11.2.4.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 11.2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 11.2.4.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{1 \pm \sqrt{1 + 4}}{2 \cdot 1}$

Step 11.2.4.1.3

Add $1$ and $4$ .

$x = \frac{1 \pm \sqrt{5}}{2 \cdot 1}$

Step 11.2.4.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm \sqrt{5}}{2}$

Step 11.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{5}}{2}$

Step 11.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 11.2.5.1

Simplify the numerator.

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Step 11.2.5.1.1

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

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

Step 11.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 11.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 11.2.5.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{1 \pm \sqrt{1 + 4}}{2 \cdot 1}$

Step 11.2.5.1.3

Add $1$ and $4$ .

$x = \frac{1 \pm \sqrt{5}}{2 \cdot 1}$

Step 11.2.5.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm \sqrt{5}}{2}$

Step 11.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{5}}{2}$

Step 11.2.6

The final answer is the combination of both solutions.

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

Step 12

The final solution is all the values that make ${(2 x + 3)}^{2} (x^{2} - x - 1) = 0$ true.

$x = - \frac{3}{2}, \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{2}, \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}$

Decimal Form:

$x = - 1.5, 1.61803398 \dots, - 0.61803398 \dots$

Mixed Number Form:

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

Step 14