Precalculus Examples

$20 x^{4} + 52 x^{3} - 45 x^{2} + 5 x + 1$

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.

$p = \pm 1$

$q = \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{5}, \pm \frac{1}{10}, \pm \frac{1}{20}$

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.

$20 {(\frac{1}{2})}^{4} + 52 {(\frac{1}{2})}^{3} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

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

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

$20 \frac{1^{4}}{2^{4}} + 52 {(\frac{1}{2})}^{3} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.2

One to any power is one.

$20 \frac{1}{2^{4}} + 52 {(\frac{1}{2})}^{3} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.3

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

$20 (\frac{1}{16}) + 52 {(\frac{1}{2})}^{3} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.4

Cancel the common factor of $4$ .

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Step 4.1.4.1

Factor $4$ out of $20$ .

$4 (5) \frac{1}{16} + 52 {(\frac{1}{2})}^{3} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.4.2

Factor $4$ out of $16$ .

$4 \cdot 5 \frac{1}{4 \cdot 4} + 52 {(\frac{1}{2})}^{3} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.4.3

Cancel the common factor.

$4 \cdot 5 \frac{1}{4 \cdot 4} + 52 {(\frac{1}{2})}^{3} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.4.4

Rewrite the expression.

$5 (\frac{1}{4}) + 52 {(\frac{1}{2})}^{3} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.5

Combine $5$ and $\frac{1}{4}$ .

$\frac{5}{4} + 52 {(\frac{1}{2})}^{3} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.6

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

$\frac{5}{4} + 52 \frac{1^{3}}{2^{3}} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.7

One to any power is one.

$\frac{5}{4} + 52 \frac{1}{2^{3}} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.8

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

$\frac{5}{4} + 52 (\frac{1}{8}) - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.9

Cancel the common factor of $4$ .

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Step 4.1.9.1

Factor $4$ out of $52$ .

$\frac{5}{4} + 4 (13) \frac{1}{8} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.9.2

Factor $4$ out of $8$ .

$\frac{5}{4} + 4 \cdot 13 \frac{1}{4 \cdot 2} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.9.3

Cancel the common factor.

$\frac{5}{4} + 4 \cdot 13 \frac{1}{4 \cdot 2} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.9.4

Rewrite the expression.

$\frac{5}{4} + 13 (\frac{1}{2}) - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.10

Combine $13$ and $\frac{1}{2}$ .

$\frac{5}{4} + \frac{13}{2} - 45 {(\frac{1}{2})}^{2} + 5 (\frac{1}{2}) + 1$

Step 4.1.11

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

$\frac{5}{4} + \frac{13}{2} - 45 \frac{1^{2}}{2^{2}} + 5 (\frac{1}{2}) + 1$

Step 4.1.12

One to any power is one.

$\frac{5}{4} + \frac{13}{2} - 45 \frac{1}{2^{2}} + 5 (\frac{1}{2}) + 1$

Step 4.1.13

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

$\frac{5}{4} + \frac{13}{2} - 45 (\frac{1}{4}) + 5 (\frac{1}{2}) + 1$

Step 4.1.14

Combine $- 45$ and $\frac{1}{4}$ .

$\frac{5}{4} + \frac{13}{2} + \frac{- 45}{4} + 5 (\frac{1}{2}) + 1$

Step 4.1.15

Move the negative in front of the fraction.

$\frac{5}{4} + \frac{13}{2} - \frac{45}{4} + 5 (\frac{1}{2}) + 1$

Step 4.1.16

Combine $5$ and $\frac{1}{2}$ .

$\frac{5}{4} + \frac{13}{2} - \frac{45}{4} + \frac{5}{2} + 1$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine the numerators over the common denominator.

$1 + \frac{5 - 45}{4} + \frac{13 + 5}{2}$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $45$ from $5$ .

$1 + \frac{- 40}{4} + \frac{13 + 5}{2}$

Step 4.2.2.2

Add $13$ and $5$ .

$1 + \frac{- 40}{4} + \frac{18}{2}$

Step 4.3

Find the common denominator.

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Step 4.3.1

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

$\frac{1}{1} + \frac{- 40}{4} + \frac{18}{2}$

Step 4.3.2

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

$\frac{1}{1} \cdot \frac{4}{4} + \frac{- 40}{4} + \frac{18}{2}$

Step 4.3.3

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

$\frac{4}{4} + \frac{- 40}{4} + \frac{18}{2}$

Step 4.3.4

Multiply $\frac{18}{2}$ by $\frac{2}{2}$ .

$\frac{4}{4} + \frac{- 40}{4} + \frac{18}{2} \cdot \frac{2}{2}$

Step 4.3.5

Multiply $\frac{18}{2}$ by $\frac{2}{2}$ .

$\frac{4}{4} + \frac{- 40}{4} + \frac{18 \cdot 2}{2 \cdot 2}$

Step 4.3.6

Multiply $2$ by $2$ .

$\frac{4}{4} + \frac{- 40}{4} + \frac{18 \cdot 2}{4}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{4 - 40 + 18 \cdot 2}{4}$

Step 4.5

Simplify the expression.

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Step 4.5.1

Multiply $18$ by $2$ .

$\frac{4 - 40 + 36}{4}$

Step 4.5.2

Subtract $40$ from $4$ .

$\frac{- 36 + 36}{4}$

Step 4.5.3

Add $- 36$ and $36$ .

$\frac{0}{4}$

Step 4.5.4

Divide $0$ by $4$ .

$0$

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.

$\frac{20 x^{4} + 52 x^{3} - 45 x^{2} + 5 x + 1}{x - \frac{1}{2}}$

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.

$\frac{1}{2}$	$20$	$52$	$- 45$	$5$	$1$

Step 6.2

The first number in the dividend $(20)$ is put into the first position of the result area (below the horizontal line).

$\frac{1}{2}$	$20$	$52$	$- 45$	$5$	$1$

	$20$

Step 6.3

Multiply the newest entry in the result $(20)$ by the divisor $(\frac{1}{2})$ and place the result of $(10)$ under the next term in the dividend $(52)$ .

$\frac{1}{2}$	$20$	$52$	$- 45$	$5$	$1$
		$10$
	$20$

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.

$\frac{1}{2}$	$20$	$52$	$- 45$	$5$	$1$
		$10$
	$20$	$62$

Step 6.5

Multiply the newest entry in the result $(62)$ by the divisor $(\frac{1}{2})$ and place the result of $(31)$ under the next term in the dividend $(- 45)$ .

$\frac{1}{2}$	$20$	$52$	$- 45$	$5$	$1$
		$10$	$31$
	$20$	$62$

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.

$\frac{1}{2}$	$20$	$52$	$- 45$	$5$	$1$
		$10$	$31$
	$20$	$62$	$- 14$

Step 6.7

Multiply the newest entry in the result $(- 14)$ by the divisor $(\frac{1}{2})$ and place the result of $(- 7)$ under the next term in the dividend $(5)$ .

$\frac{1}{2}$	$20$	$52$	$- 45$	$5$	$1$
		$10$	$31$	$- 7$
	$20$	$62$	$- 14$

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.

$\frac{1}{2}$	$20$	$52$	$- 45$	$5$	$1$
		$10$	$31$	$- 7$
	$20$	$62$	$- 14$	$- 2$

Step 6.9

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 $(1)$ .

$\frac{1}{2}$	$20$	$52$	$- 45$	$5$	$1$
		$10$	$31$	$- 7$	$- 1$
	$20$	$62$	$- 14$	$- 2$

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.

$\frac{1}{2}$	$20$	$52$	$- 45$	$5$	$1$
		$10$	$31$	$- 7$	$- 1$
	$20$	$62$	$- 14$	$- 2$	$0$

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.

$20 x^{3} + 62 x^{2} + (- 14) x - 2$

Step 6.12

Simplify the quotient polynomial.

$20 x^{3} + 62 x^{2} - 14 x - 2$

Step 7

Factor $2$ out of $20 x^{3} + 62 x^{2} - 14 x - 2$ .

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Step 7.1

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

$(x - \frac{1}{2}) (2 (10 x^{3}) + 62 x^{2} - 14 x - 2)$

Step 7.2

Factor $2$ out of $62 x^{2}$ .

$(x - \frac{1}{2}) (2 (10 x^{3}) + 2 (31 x^{2}) - 14 x - 2)$

Step 7.3

Factor $2$ out of $- 14 x$ .

$(x - \frac{1}{2}) (2 (10 x^{3}) + 2 (31 x^{2}) + 2 (- 7 x) - 2)$

Step 7.4

Factor $2$ out of $- 2$ .

$(x - \frac{1}{2}) (2 (10 x^{3}) + 2 (31 x^{2}) + 2 (- 7 x) + 2 (- 1))$

Step 7.5

Factor $2$ out of $2 (10 x^{3}) + 2 (31 x^{2})$ .

$(x - \frac{1}{2}) (2 (10 x^{3} + 31 x^{2}) + 2 (- 7 x) + 2 (- 1))$

Step 7.6

Factor $2$ out of $2 (10 x^{3} + 31 x^{2}) + 2 (- 7 x)$ .

$(x - \frac{1}{2}) (2 (10 x^{3} + 31 x^{2} - 7 x) + 2 (- 1))$

Step 7.7

Factor $2$ out of $2 (10 x^{3} + 31 x^{2} - 7 x) + 2 (- 1)$ .

$(x - \frac{1}{2}) (2 (10 x^{3} + 31 x^{2} - 7 x - 1))$

Step 8

Factor the left side of the equation.

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Step 8.1

Factor $20 x^{4} + 52 x^{3} - 45 x^{2} + 5 x + 1$ 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.

$p = \pm 1$

$q = \pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

Step 8.1.2

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

$\pm 1, \pm 0.05, \pm 0.5, \pm 0.1, \pm 0.25, \pm 0.2$

Step 8.1.3

Substitute $- 0.1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.1$ is a root of the polynomial.

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Step 8.1.3.1

Substitute $- 0.1$ into the polynomial.

$20 {(- 0.1)}^{4} + 52 {(- 0.1)}^{3} - 45 {(- 0.1)}^{2} + 5 \cdot - 0.1 + 1$

Step 8.1.3.2

Raise $- 0.1$ to the power of $4$ .

$20 \cdot 0.0001 + 52 {(- 0.1)}^{3} - 45 {(- 0.1)}^{2} + 5 \cdot - 0.1 + 1$

Step 8.1.3.3

Multiply $20$ by $0.0001$ .

$0.002 + 52 {(- 0.1)}^{3} - 45 {(- 0.1)}^{2} + 5 \cdot - 0.1 + 1$

Step 8.1.3.4

Raise $- 0.1$ to the power of $3$ .

$0.002 + 52 \cdot - 0.001 - 45 {(- 0.1)}^{2} + 5 \cdot - 0.1 + 1$

Step 8.1.3.5

Multiply $52$ by $- 0.001$ .

$0.002 - 0.052 - 45 {(- 0.1)}^{2} + 5 \cdot - 0.1 + 1$

Step 8.1.3.6

Subtract $0.052$ from $0.002$ .

$- 0.05 - 45 {(- 0.1)}^{2} + 5 \cdot - 0.1 + 1$

Step 8.1.3.7

Raise $- 0.1$ to the power of $2$ .

$- 0.05 - 45 \cdot 0.01 + 5 \cdot - 0.1 + 1$

Step 8.1.3.8

Multiply $- 45$ by $0.01$ .

$- 0.05 - 0.45 + 5 \cdot - 0.1 + 1$

Step 8.1.3.9

Subtract $0.45$ from $- 0.05$ .

$- 0.5 + 5 \cdot - 0.1 + 1$

Step 8.1.3.10

Multiply $5$ by $- 0.1$ .

$- 0.5 - 0.5 + 1$

Step 8.1.3.11

Subtract $0.5$ from $- 0.5$ .

$- 1 + 1$

Step 8.1.3.12

Add $- 1$ and $1$ .

$0$

Step 8.1.4

Since $- 0.1$ is a known root, divide the polynomial by $10 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{20 x^{4} + 52 x^{3} - 45 x^{2} + 5 x + 1}{10 x + 1}$

Step 8.1.5

Divide $20 x^{4} + 52 x^{3} - 45 x^{2} + 5 x + 1$ by $10 x + 1$ .

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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 $0$ .

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

Step 8.1.5.2

Divide the highest order term in the dividend $20 x^{4}$ by the highest order term in divisor $10 x$ .

$2 x^{3}$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

Step 8.1.5.3

Multiply the new quotient term by the divisor.

$2 x^{3}$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

Step 8.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $20 x^{4} + 2 x^{3}$

$2 x^{3}$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

Step 8.1.5.5

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

$2 x^{3}$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

Step 8.1.5.6

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

$2 x^{3}$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

Step 8.1.5.7

Divide the highest order term in the dividend $50 x^{3}$ by the highest order term in divisor $10 x$ .

$2 x^{3}$

$5 x^{2}$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

Step 8.1.5.8

Multiply the new quotient term by the divisor.

$2 x^{3}$

$5 x^{2}$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

$50 x^{3}$

$5 x^{2}$

Step 8.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $50 x^{3} + 5 x^{2}$

$2 x^{3}$

$5 x^{2}$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

$50 x^{3}$

$5 x^{2}$

Step 8.1.5.10

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

$2 x^{3}$

$5 x^{2}$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

$50 x^{3}$

$5 x^{2}$

$50 x^{2}$

Step 8.1.5.11

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

$2 x^{3}$

$5 x^{2}$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

$50 x^{3}$

$5 x^{2}$

$50 x^{2}$

$5 x$

Step 8.1.5.12

Divide the highest order term in the dividend $- 50 x^{2}$ by the highest order term in divisor $10 x$ .

$2 x^{3}$

$5 x^{2}$

$5 x$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

$50 x^{3}$

$5 x^{2}$

$50 x^{2}$

$5 x$

Step 8.1.5.13

Multiply the new quotient term by the divisor.

$2 x^{3}$

$5 x^{2}$

$5 x$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

$50 x^{3}$

$5 x^{2}$

$50 x^{2}$

$5 x$

$50 x^{2}$

$5 x$

Step 8.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 50 x^{2} - 5 x$

$2 x^{3}$

$5 x^{2}$

$5 x$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

$50 x^{3}$

$5 x^{2}$

$50 x^{2}$

$5 x$

$50 x^{2}$

$5 x$

Step 8.1.5.15

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

$2 x^{3}$

$5 x^{2}$

$5 x$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

$50 x^{3}$

$5 x^{2}$

$50 x^{2}$

$5 x$

$50 x^{2}$

$5 x$

$10 x$

Step 8.1.5.16

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

$2 x^{3}$

$5 x^{2}$

$5 x$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

$50 x^{3}$

$5 x^{2}$

$50 x^{2}$

$5 x$

$50 x^{2}$

$5 x$

$10 x$

$1$

Step 8.1.5.17

Divide the highest order term in the dividend $10 x$ by the highest order term in divisor $10 x$ .

$2 x^{3}$

$5 x^{2}$

$5 x$

$1$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

$50 x^{3}$

$5 x^{2}$

$50 x^{2}$

$5 x$

$50 x^{2}$

$5 x$

$10 x$

$1$

Step 8.1.5.18

Multiply the new quotient term by the divisor.

$2 x^{3}$

$5 x^{2}$

$5 x$

$1$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

$50 x^{3}$

$5 x^{2}$

$50 x^{2}$

$5 x$

$50 x^{2}$

$5 x$

$10 x$

$1$

$10 x$

$1$

Step 8.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $10 x + 1$

$2 x^{3}$

$5 x^{2}$

$5 x$

$1$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

$50 x^{3}$

$5 x^{2}$

$50 x^{2}$

$5 x$

$50 x^{2}$

$5 x$

$10 x$

$1$

$10 x$

$1$

Step 8.1.5.20

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

$2 x^{3}$

$5 x^{2}$

$5 x$

$1$

$10 x$

$1$

$20 x^{4}$

$52 x^{3}$

$45 x^{2}$

$5 x$

$1$

$20 x^{4}$

$2 x^{3}$

$50 x^{3}$

$45 x^{2}$

$50 x^{3}$

$5 x^{2}$

$50 x^{2}$

$5 x$

$50 x^{2}$

$5 x$

$10 x$

$1$

$10 x$

$1$

$0$

Step 8.1.5.21

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

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

Step 8.1.6

Write $20 x^{4} + 52 x^{3} - 45 x^{2} + 5 x + 1$ as a set of factors.

$(10 x + 1) (2 x^{3} + 5 x^{2} - 5 x + 1) = 0$

Step 8.2

Factor $2 x^{3} + 5 x^{2} - 5 x + 1$ using the rational roots test.

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Step 8.2.1

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

$p = \pm 1$

$q = \pm 1, \pm 2$

Step 8.2.1.2

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

$\pm 1, \pm 0.5$

Step 8.2.1.3

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 8.2.1.3.1

Substitute $0.5$ into the polynomial.

$2 \cdot {0.5}^{3} + 5 \cdot {0.5}^{2} - 5 \cdot 0.5 + 1$

Step 8.2.1.3.2

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

$2 \cdot 0.125 + 5 \cdot {0.5}^{2} - 5 \cdot 0.5 + 1$

Step 8.2.1.3.3

Multiply $2$ by $0.125$ .

$0.25 + 5 \cdot {0.5}^{2} - 5 \cdot 0.5 + 1$

Step 8.2.1.3.4

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

$0.25 + 5 \cdot 0.25 - 5 \cdot 0.5 + 1$

Step 8.2.1.3.5

Multiply $5$ by $0.25$ .

$0.25 + 1.25 - 5 \cdot 0.5 + 1$

Step 8.2.1.3.6

Add $0.25$ and $1.25$ .

$1.5 - 5 \cdot 0.5 + 1$

Step 8.2.1.3.7

Multiply $- 5$ by $0.5$ .

$1.5 - 2.5 + 1$

Step 8.2.1.3.8

Subtract $2.5$ from $1.5$ .

$- 1 + 1$

Step 8.2.1.3.9

Add $- 1$ and $1$ .

$0$

Step 8.2.1.4

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.

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

Step 8.2.1.5

Divide $2 x^{3} + 5 x^{2} - 5 x + 1$ by $2 x - 1$ .

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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 $0$ .

$2 x$

$1$

$2 x^{3}$

$5 x^{2}$

$5 x$

$1$

Step 8.2.1.5.2

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

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

Step 8.2.1.5.3

Multiply the new quotient term by the divisor.

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

Step 8.2.1.5.4

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

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

Step 8.2.1.5.5

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

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$5 x$	+	$1$
			-	$2 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$

Step 8.2.1.5.6

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

				$x^{2}$
$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$5 x$	+	$1$
			-	$2 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$5 x$

Step 8.2.1.5.7

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

				$x^{2}$	+	$3 x$
$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$5 x$	+	$1$
			-	$2 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$5 x$

Step 8.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$
$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$5 x$	+	$1$
			-	$2 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$5 x$
					+	$6 x^{2}$	-	$3 x$

Step 8.2.1.5.9

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

				$x^{2}$	+	$3 x$
$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$5 x$	+	$1$
			-	$2 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$5 x$
					-	$6 x^{2}$	+	$3 x$

Step 8.2.1.5.10

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

				$x^{2}$	+	$3 x$
$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$5 x$	+	$1$
			-	$2 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$5 x$
					-	$6 x^{2}$	+	$3 x$
							-	$2 x$

Step 8.2.1.5.11

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

				$x^{2}$	+	$3 x$
$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$5 x$	+	$1$
			-	$2 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$5 x$
					-	$6 x^{2}$	+	$3 x$
							-	$2 x$	+	$1$

Step 8.2.1.5.12

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

				$x^{2}$	+	$3 x$	-	$1$
$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$5 x$	+	$1$
			-	$2 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$5 x$
					-	$6 x^{2}$	+	$3 x$
							-	$2 x$	+	$1$

Step 8.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$	-	$1$
$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$5 x$	+	$1$
			-	$2 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$5 x$
					-	$6 x^{2}$	+	$3 x$
							-	$2 x$	+	$1$
							-	$2 x$	+	$1$

Step 8.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x + 1$

				$x^{2}$	+	$3 x$	-	$1$
$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$5 x$	+	$1$
			-	$2 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$5 x$
					-	$6 x^{2}$	+	$3 x$
							-	$2 x$	+	$1$
							+	$2 x$	-	$1$

Step 8.2.1.5.15

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

				$x^{2}$	+	$3 x$	-	$1$
$2 x$	-	$1$		$2 x^{3}$	+	$5 x^{2}$	-	$5 x$	+	$1$
			-	$2 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$5 x$
					-	$6 x^{2}$	+	$3 x$
							-	$2 x$	+	$1$
							+	$2 x$	-	$1$
										$0$

Step 8.2.1.5.16

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

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

Step 8.2.1.6

Write $2 x^{3} + 5 x^{2} - 5 x + 1$ as a set of factors.

$(10 x + 1) ((2 x - 1) (x^{2} + 3 x - 1)) = 0$

Step 8.2.2

Remove unnecessary parentheses.

$(10 x + 1) (2 x - 1) (x^{2} + 3 x - 1) = 0$

Step 9

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

$10 x + 1 = 0$

$2 x - 1 = 0$

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

Step 10

Set $10 x + 1$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $10 x + 1$ equal to $0$ .

$10 x + 1 = 0$

Step 10.2

Solve $10 x + 1 = 0$ for $x$ .

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Step 10.2.1

Subtract $1$ from both sides of the equation.

$10 x = - 1$

Step 10.2.2

Divide each term in $10 x = - 1$ by $10$ and simplify.

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Step 10.2.2.1

Divide each term in $10 x = - 1$ by $10$ .

$\frac{10 x}{10} = \frac{- 1}{10}$

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of $10$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{- 1}{10}$

Step 10.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{10}$

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.

$x = - \frac{1}{10}$

Step 11

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

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Step 11.1

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

$2 x - 1 = 0$

Step 11.2

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

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Step 11.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 11.2.2

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

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Step 11.2.2.1

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

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

Step 11.2.2.2

Simplify the left side.

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Step 11.2.2.2.1

Cancel the common factor of $2$ .

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Step 11.2.2.2.1.1

Cancel the common factor.

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

Step 11.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 12

Set $x^{2} + 3 x - 1$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x^{2} + 3 x - 1$ equal to $0$ .

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

Step 12.2

Solve $x^{2} + 3 x - 1 = 0$ for $x$ .

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Step 12.2.1

Use the quadratic formula to find the solutions.

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

Step 12.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 12.2.3

Simplify.

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Step 12.2.3.1

Simplify the numerator.

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Step 12.2.3.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 12.2.3.1.2

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

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Step 12.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 1}}{2 \cdot 1}$

Step 12.2.3.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 + 4}}{2 \cdot 1}$

Step 12.2.3.1.3

Add $9$ and $4$ .

$x = \frac{- 3 \pm \sqrt{13}}{2 \cdot 1}$

Step 12.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{13}}{2}$

Step 12.2.4

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

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Step 12.2.4.1

Simplify the numerator.

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Step 12.2.4.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 12.2.4.1.2

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

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Step 12.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 1}}{2 \cdot 1}$

Step 12.2.4.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 + 4}}{2 \cdot 1}$

Step 12.2.4.1.3

Add $9$ and $4$ .

$x = \frac{- 3 \pm \sqrt{13}}{2 \cdot 1}$

Step 12.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{13}}{2}$

Step 12.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + \sqrt{13}}{2}$

Step 12.2.4.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + \sqrt{13}}{2}$

Step 12.2.4.5

Factor $- 1$ out of $\sqrt{13}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{13})}{2}$

Step 12.2.4.6

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{13})$ .

$x = \frac{- 1 (3 - \sqrt{13})}{2}$

Step 12.2.4.7

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{13}}{2}$

Step 12.2.5

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

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Step 12.2.5.1

Simplify the numerator.

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Step 12.2.5.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 12.2.5.1.2

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

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Step 12.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 1}}{2 \cdot 1}$

Step 12.2.5.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 + 4}}{2 \cdot 1}$

Step 12.2.5.1.3

Add $9$ and $4$ .

$x = \frac{- 3 \pm \sqrt{13}}{2 \cdot 1}$

Step 12.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{13}}{2}$

Step 12.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - \sqrt{13}}{2}$

Step 12.2.5.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - \sqrt{13}}{2}$

Step 12.2.5.5

Factor $- 1$ out of $- \sqrt{13}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{13})}{2}$

Step 12.2.5.6

Factor $- 1$ out of $- 1 (3) - (\sqrt{13})$ .

$x = \frac{- 1 (3 + \sqrt{13})}{2}$

Step 12.2.5.7

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{13}}{2}$

Step 12.2.6

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{13}}{2}, - \frac{3 + \sqrt{13}}{2}$

Step 13

The final solution is all the values that make $(10 x + 1) (2 x - 1) (x^{2} + 3 x - 1) = 0$ true.

$x = - \frac{1}{10}, \frac{1}{2}, - \frac{3 - \sqrt{13}}{2}, - \frac{3 + \sqrt{13}}{2}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{10}, \frac{1}{2}, - \frac{3 - \sqrt{13}}{2}, - \frac{3 + \sqrt{13}}{2}$

Decimal Form:

$x = - 0.1, 0.5, 0.30277563 \dots, - 3.30277563 \dots$

Step 15