Precalculus Examples

$4 x^{4} + 17 x^{3} - 11 x^{2} + 17 x - 15$

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.

$p = \pm 1, \pm 3, \pm 5, \pm 15$

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

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{4}, \pm 15, \pm \frac{15}{2}, \pm \frac{15}{4}$

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.

$4 {(\frac{3}{4})}^{4} + 17 {(\frac{3}{4})}^{3} - 11 {(\frac{3}{4})}^{2} + 17 (\frac{3}{4}) - 15$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = \frac{3}{4}$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the product rule to $\frac{3}{4}$ .

$4 \frac{3^{4}}{4^{4}} + 17 {(\frac{3}{4})}^{3} - 11 {(\frac{3}{4})}^{2} + 17 (\frac{3}{4}) - 15$

Step 4.1.2

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

$4 \frac{81}{4^{4}} + 17 {(\frac{3}{4})}^{3} - 11 {(\frac{3}{4})}^{2} + 17 (\frac{3}{4}) - 15$

Step 4.1.3

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

$4 (\frac{81}{256}) + 17 {(\frac{3}{4})}^{3} - 11 {(\frac{3}{4})}^{2} + 17 (\frac{3}{4}) - 15$

Step 4.1.4

Cancel the common factor of $4$ .

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Step 4.1.4.1

Factor $4$ out of $256$ .

$4 \frac{81}{4 (64)} + 17 {(\frac{3}{4})}^{3} - 11 {(\frac{3}{4})}^{2} + 17 (\frac{3}{4}) - 15$

Step 4.1.4.2

Cancel the common factor.

$4 \frac{81}{4 \cdot 64} + 17 {(\frac{3}{4})}^{3} - 11 {(\frac{3}{4})}^{2} + 17 (\frac{3}{4}) - 15$

Step 4.1.4.3

Rewrite the expression.

$\frac{81}{64} + 17 {(\frac{3}{4})}^{3} - 11 {(\frac{3}{4})}^{2} + 17 (\frac{3}{4}) - 15$

Step 4.1.5

Apply the product rule to $\frac{3}{4}$ .

$\frac{81}{64} + 17 \frac{3^{3}}{4^{3}} - 11 {(\frac{3}{4})}^{2} + 17 (\frac{3}{4}) - 15$

Step 4.1.6

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

$\frac{81}{64} + 17 \frac{27}{4^{3}} - 11 {(\frac{3}{4})}^{2} + 17 (\frac{3}{4}) - 15$

Step 4.1.7

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

$\frac{81}{64} + 17 (\frac{27}{64}) - 11 {(\frac{3}{4})}^{2} + 17 (\frac{3}{4}) - 15$

Step 4.1.8

Multiply $17 (\frac{27}{64})$ .

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Step 4.1.8.1

Combine $17$ and $\frac{27}{64}$ .

$\frac{81}{64} + \frac{17 \cdot 27}{64} - 11 {(\frac{3}{4})}^{2} + 17 (\frac{3}{4}) - 15$

Step 4.1.8.2

Multiply $17$ by $27$ .

$\frac{81}{64} + \frac{459}{64} - 11 {(\frac{3}{4})}^{2} + 17 (\frac{3}{4}) - 15$

Step 4.1.9

Apply the product rule to $\frac{3}{4}$ .

$\frac{81}{64} + \frac{459}{64} - 11 \frac{3^{2}}{4^{2}} + 17 (\frac{3}{4}) - 15$

Step 4.1.10

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

$\frac{81}{64} + \frac{459}{64} - 11 \frac{9}{4^{2}} + 17 (\frac{3}{4}) - 15$

Step 4.1.11

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

$\frac{81}{64} + \frac{459}{64} - 11 (\frac{9}{16}) + 17 (\frac{3}{4}) - 15$

Step 4.1.12

Multiply $- 11 (\frac{9}{16})$ .

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Step 4.1.12.1

Combine $- 11$ and $\frac{9}{16}$ .

$\frac{81}{64} + \frac{459}{64} + \frac{- 11 \cdot 9}{16} + 17 (\frac{3}{4}) - 15$

Step 4.1.12.2

Multiply $- 11$ by $9$ .

$\frac{81}{64} + \frac{459}{64} + \frac{- 99}{16} + 17 (\frac{3}{4}) - 15$

Step 4.1.13

Move the negative in front of the fraction.

$\frac{81}{64} + \frac{459}{64} - \frac{99}{16} + 17 (\frac{3}{4}) - 15$

Step 4.1.14

Multiply $17 (\frac{3}{4})$ .

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Step 4.1.14.1

Combine $17$ and $\frac{3}{4}$ .

$\frac{81}{64} + \frac{459}{64} - \frac{99}{16} + \frac{17 \cdot 3}{4} - 15$

Step 4.1.14.2

Multiply $17$ by $3$ .

$\frac{81}{64} + \frac{459}{64} - \frac{99}{16} + \frac{51}{4} - 15$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine the numerators over the common denominator.

$- 15 + \frac{81 + 459}{64} + \frac{- 99}{16} + \frac{51}{4}$

Step 4.2.2

Add $81$ and $459$ .

$- 15 + \frac{540}{64} + \frac{- 99}{16} + \frac{51}{4}$

Step 4.3

Find the common denominator.

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Step 4.3.1

Write $- 15$ as a fraction with denominator $1$ .

$\frac{- 15}{1} + \frac{540}{64} + \frac{- 99}{16} + \frac{51}{4}$

Step 4.3.2

Multiply $\frac{- 15}{1}$ by $\frac{64}{64}$ .

$\frac{- 15}{1} \cdot \frac{64}{64} + \frac{540}{64} + \frac{- 99}{16} + \frac{51}{4}$

Step 4.3.3

Multiply $\frac{- 15}{1}$ by $\frac{64}{64}$ .

$\frac{- 15 \cdot 64}{64} + \frac{540}{64} + \frac{- 99}{16} + \frac{51}{4}$

Step 4.3.4

Multiply $\frac{- 99}{16}$ by $\frac{4}{4}$ .

$\frac{- 15 \cdot 64}{64} + \frac{540}{64} + \frac{- 99}{16} \cdot \frac{4}{4} + \frac{51}{4}$

Step 4.3.5

Multiply $\frac{- 99}{16}$ by $\frac{4}{4}$ .

$\frac{- 15 \cdot 64}{64} + \frac{540}{64} + \frac{- 99 \cdot 4}{16 \cdot 4} + \frac{51}{4}$

Step 4.3.6

Multiply $\frac{51}{4}$ by $\frac{16}{16}$ .

$\frac{- 15 \cdot 64}{64} + \frac{540}{64} + \frac{- 99 \cdot 4}{16 \cdot 4} + \frac{51}{4} \cdot \frac{16}{16}$

Step 4.3.7

Multiply $\frac{51}{4}$ by $\frac{16}{16}$ .

$\frac{- 15 \cdot 64}{64} + \frac{540}{64} + \frac{- 99 \cdot 4}{16 \cdot 4} + \frac{51 \cdot 16}{4 \cdot 16}$

Step 4.3.8

Reorder the factors of $16 \cdot 4$ .

$\frac{- 15 \cdot 64}{64} + \frac{540}{64} + \frac{- 99 \cdot 4}{4 \cdot 16} + \frac{51 \cdot 16}{4 \cdot 16}$

Step 4.3.9

Multiply $4$ by $16$ .

$\frac{- 15 \cdot 64}{64} + \frac{540}{64} + \frac{- 99 \cdot 4}{64} + \frac{51 \cdot 16}{4 \cdot 16}$

Step 4.3.10

Multiply $4$ by $16$ .

$\frac{- 15 \cdot 64}{64} + \frac{540}{64} + \frac{- 99 \cdot 4}{64} + \frac{51 \cdot 16}{64}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{- 15 \cdot 64 + 540 - 99 \cdot 4 + 51 \cdot 16}{64}$

Step 4.5

Simplify each term.

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Step 4.5.1

Multiply $- 15$ by $64$ .

$\frac{- 960 + 540 - 99 \cdot 4 + 51 \cdot 16}{64}$

Step 4.5.2

Multiply $- 99$ by $4$ .

$\frac{- 960 + 540 - 396 + 51 \cdot 16}{64}$

Step 4.5.3

Multiply $51$ by $16$ .

$\frac{- 960 + 540 - 396 + 816}{64}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $- 960$ and $540$ .

$\frac{- 420 - 396 + 816}{64}$

Step 4.6.2

Subtract $396$ from $- 420$ .

$\frac{- 816 + 816}{64}$

Step 4.6.3

Add $- 816$ and $816$ .

$\frac{0}{64}$

Step 4.6.4

Divide $0$ by $64$ .

$0$

Step 5

Since $\frac{3}{4}$ is a known root, divide the polynomial by $x - \frac{3}{4}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{4} + 17 x^{3} - 11 x^{2} + 17 x - 15}{x - \frac{3}{4}}$

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.

$\frac{3}{4}$	$4$	$17$	$- 11$	$17$	$- 15$

Step 6.2

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

$\frac{3}{4}$	$4$	$17$	$- 11$	$17$	$- 15$

	$4$

Step 6.3

Multiply the newest entry in the result $(4)$ by the divisor $(\frac{3}{4})$ and place the result of $(3)$ under the next term in the dividend $(17)$ .

$\frac{3}{4}$	$4$	$17$	$- 11$	$17$	$- 15$
		$3$
	$4$

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.

$\frac{3}{4}$	$4$	$17$	$- 11$	$17$	$- 15$
		$3$
	$4$	$20$

Step 6.5

Multiply the newest entry in the result $(20)$ by the divisor $(\frac{3}{4})$ and place the result of $(15)$ under the next term in the dividend $(- 11)$ .

$\frac{3}{4}$	$4$	$17$	$- 11$	$17$	$- 15$
		$3$	$15$
	$4$	$20$

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.

$\frac{3}{4}$	$4$	$17$	$- 11$	$17$	$- 15$
		$3$	$15$
	$4$	$20$	$4$

Step 6.7

Multiply the newest entry in the result $(4)$ by the divisor $(\frac{3}{4})$ and place the result of $(3)$ under the next term in the dividend $(17)$ .

$\frac{3}{4}$	$4$	$17$	$- 11$	$17$	$- 15$
		$3$	$15$	$3$
	$4$	$20$	$4$

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.

$\frac{3}{4}$	$4$	$17$	$- 11$	$17$	$- 15$
		$3$	$15$	$3$
	$4$	$20$	$4$	$20$

Step 6.9

Multiply the newest entry in the result $(20)$ by the divisor $(\frac{3}{4})$ and place the result of $(15)$ under the next term in the dividend $(- 15)$ .

$\frac{3}{4}$	$4$	$17$	$- 11$	$17$	$- 15$
		$3$	$15$	$3$	$15$
	$4$	$20$	$4$	$20$

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.

$\frac{3}{4}$	$4$	$17$	$- 11$	$17$	$- 15$
		$3$	$15$	$3$	$15$
	$4$	$20$	$4$	$20$	$0$

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.

$4 x^{3} + 20 x^{2} + (4) x + 20$

Step 6.12

Simplify the quotient polynomial.

$4 x^{3} + 20 x^{2} + 4 x + 20$

Step 7

Factor $4$ out of $4 x^{3} + 20 x^{2} + 4 x + 20$ .

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Step 7.1

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

$(x - \frac{3}{4}) (4 (x^{3}) + 20 x^{2} + 4 x + 20)$

Step 7.2

Factor $4$ out of $20 x^{2}$ .

$(x - \frac{3}{4}) (4 (x^{3}) + 4 (5 x^{2}) + 4 x + 20)$

Step 7.3

Factor $4$ out of $4 x$ .

$(x - \frac{3}{4}) (4 (x^{3}) + 4 (5 x^{2}) + 4 (x) + 20)$

Step 7.4

Factor $4$ out of $20$ .

$(x - \frac{3}{4}) (4 (x^{3}) + 4 (5 x^{2}) + 4 x + 4 \cdot 5)$

Step 7.5

Factor $4$ out of $4 (x^{3}) + 4 (5 x^{2})$ .

$(x - \frac{3}{4}) (4 (x^{3} + 5 x^{2}) + 4 x + 4 \cdot 5)$

Step 7.6

Factor $4$ out of $4 (x^{3} + 5 x^{2}) + 4 x$ .

$(x - \frac{3}{4}) (4 (x^{3} + 5 x^{2} + x) + 4 \cdot 5)$

Step 7.7

Factor $4$ out of $4 (x^{3} + 5 x^{2} + x) + 4 \cdot 5$ .

$(x - \frac{3}{4}) (4 (x^{3} + 5 x^{2} + x + 5))$

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.

$(x - \frac{3}{4}) (4 ((x^{3} + 5 x^{2}) + x + 5))$

Step 8.2

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

$(x - \frac{3}{4}) (4 (x^{2} (x + 5) + 1 (x + 5)))$

Step 9

Factor.

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Step 9.1

Factor the polynomial by factoring out the greatest common factor, $x + 5$ .

$(x - \frac{3}{4}) (4 ((x + 5) (x^{2} + 1)))$

Step 9.2

Remove unnecessary parentheses.

$(x - \frac{3}{4}) (4 (x + 5) (x^{2} + 1))$

Step 10

Factor the left side of the equation.

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Step 10.1

Regroup terms.

$17 x^{3} + 17 x + 4 x^{4} - 11 x^{2} - 15 = 0$

Step 10.2

Factor $17 x$ out of $17 x^{3} + 17 x$ .

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Step 10.2.1

Factor $17 x$ out of $17 x^{3}$ .

$17 x (x^{2}) + 17 x + 4 x^{4} - 11 x^{2} - 15 = 0$

Step 10.2.2

Factor $17 x$ out of $17 x$ .

$17 x (x^{2}) + 17 x (1) + 4 x^{4} - 11 x^{2} - 15 = 0$

Step 10.2.3

Factor $17 x$ out of $17 x (x^{2}) + 17 x (1)$ .

$17 x (x^{2} + 1) + 4 x^{4} - 11 x^{2} - 15 = 0$

Step 10.3

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

$17 x (x^{2} + 1) + 4 {(x^{2})}^{2} - 11 x^{2} - 15 = 0$

Step 10.4

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

$17 x (x^{2} + 1) + 4 u^{2} - 11 u - 15 = 0$

Step 10.5

Factor by grouping.

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Step 10.5.1

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 $a \cdot c = 4 \cdot - 15 = - 60$ and whose sum is $b = - 11$ .

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Step 10.5.1.1

Factor $- 11$ out of $- 11 u$ .

$17 x (x^{2} + 1) + 4 u^{2} - 11 u - 15 = 0$

Step 10.5.1.2

Rewrite $- 11$ as $4$ plus $- 15$

$17 x (x^{2} + 1) + 4 u^{2} + (4 - 15) u - 15 = 0$

Step 10.5.1.3

Apply the distributive property.

$17 x (x^{2} + 1) + 4 u^{2} + 4 u - 15 u - 15 = 0$

Step 10.5.2

Factor out the greatest common factor from each group.

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Step 10.5.2.1

Group the first two terms and the last two terms.

$17 x (x^{2} + 1) + 4 u^{2} + 4 u - 15 u - 15 = 0$

Step 10.5.2.2

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

$17 x (x^{2} + 1) + 4 u (u + 1) - 15 (u + 1) = 0$

Step 10.5.3

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

$17 x (x^{2} + 1) + (u + 1) (4 u - 15) = 0$

Step 10.6

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

$17 x (x^{2} + 1) + (x^{2} + 1) (4 x^{2} - 15) = 0$

Step 10.7

Factor $x^{2} + 1$ out of $17 x (x^{2} + 1) + (x^{2} + 1) (4 x^{2} - 15)$ .

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Step 10.7.1

Factor $x^{2} + 1$ out of $17 x (x^{2} + 1)$ .

$(x^{2} + 1) (17 x) + (x^{2} + 1) (4 x^{2} - 15) = 0$

Step 10.7.2

Factor $x^{2} + 1$ out of $(x^{2} + 1) (17 x) + (x^{2} + 1) (4 x^{2} - 15)$ .

$(x^{2} + 1) (17 x + 4 x^{2} - 15) = 0$

Step 10.8

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(x^{2} + 1) (17 u + 4 u^{2} - 15) = 0$

Step 10.9

Factor by grouping.

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Step 10.9.1

Reorder terms.

$(x^{2} + 1) (4 u^{2} + 17 u - 15) = 0$

Step 10.9.2

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 $a \cdot c = 4 \cdot - 15 = - 60$ and whose sum is $b = 17$ .

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Step 10.9.2.1

Factor $17$ out of $17 u$ .

$(x^{2} + 1) (4 u^{2} + 17 (u) - 15) = 0$

Step 10.9.2.2

Rewrite $17$ as $- 3$ plus $20$

$(x^{2} + 1) (4 u^{2} + (- 3 + 20) u - 15) = 0$

Step 10.9.2.3

Apply the distributive property.

$(x^{2} + 1) (4 u^{2} - 3 u + 20 u - 15) = 0$

Step 10.9.3

Factor out the greatest common factor from each group.

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Step 10.9.3.1

Group the first two terms and the last two terms.

$(x^{2} + 1) ((4 u^{2} - 3 u) + 20 u - 15) = 0$

Step 10.9.3.2

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

$(x^{2} + 1) (u (4 u - 3) + 5 (4 u - 3)) = 0$

Step 10.9.4

Factor the polynomial by factoring out the greatest common factor, $4 u - 3$ .

$(x^{2} + 1) ((4 u - 3) (u + 5)) = 0$

Step 10.10

Factor.

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Step 10.10.1

Replace all occurrences of $u$ with $x$ .

$(x^{2} + 1) ((4 x - 3) (x + 5)) = 0$

Step 10.10.2

Remove unnecessary parentheses.

$(x^{2} + 1) (4 x - 3) (x + 5) = 0$

Step 11

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

$x^{2} + 1 = 0$

$4 x - 3 = 0$

$x + 5 = 0$

Step 12

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

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Step 12.1

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

$x^{2} + 1 = 0$

Step 12.2

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

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Step 12.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 12.2.2

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

$x = \pm \sqrt{- 1}$

Step 12.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 12.2.4

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

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Step 12.2.4.1

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

$x = i$

Step 12.2.4.2

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

$x = - i$

Step 12.2.4.3

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

$x = i, - i$

Step 13

Set $4 x - 3$ equal to $0$ and solve for $x$ .

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Step 13.1

Set $4 x - 3$ equal to $0$ .

$4 x - 3 = 0$

Step 13.2

Solve $4 x - 3 = 0$ for $x$ .

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Step 13.2.1

Add $3$ to both sides of the equation.

$4 x = 3$

Step 13.2.2

Divide each term in $4 x = 3$ by $4$ and simplify.

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Step 13.2.2.1

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Step 13.2.2.2

Simplify the left side.

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Step 13.2.2.2.1

Cancel the common factor of $4$ .

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Step 13.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Step 13.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 14

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 14.1

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

$x + 5 = 0$

Step 14.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 15

The final solution is all the values that make $(x^{2} + 1) (4 x - 3) (x + 5) = 0$ true.

$x = i, - i, \frac{3}{4}, - 5$

Step 16