Algebra Examples

$S (r) = r^{5} + 4 r^{4} - 13 r^{3} - 10 r^{2} - 12 r + 72$

Step 1

Set $r^{5} + 4 r^{4} - 13 r^{3} - 10 r^{2} - 12 r + 72$ equal to $0$ .

$r^{5} + 4 r^{4} - 13 r^{3} - 10 r^{2} - 12 r + 72 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Regroup terms.

$r^{5} - 13 r^{3} + 72 + 4 r^{4} - 10 r^{2} - 12 r = 0$

Step 2.1.2

Factor $r^{5} - 13 r^{3} + 72$ using the rational roots test.

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Step 2.1.2.1

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.

$p = \pm 1, \pm 72, \pm 2, \pm 36, \pm 3, \pm 24, \pm 4, \pm 18, \pm 6, \pm 12, \pm 8, \pm 9$

$q = \pm 1$

Step 2.1.2.2

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

$\pm 1, \pm 72, \pm 2, \pm 36, \pm 3, \pm 24, \pm 4, \pm 18, \pm 6, \pm 12, \pm 8, \pm 9$

Step 2.1.2.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 2.1.2.3.1

Substitute $2$ into the polynomial.

$2^{5} - 13 \cdot 2^{3} + 72$

Step 2.1.2.3.2

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

$32 - 13 \cdot 2^{3} + 72$

Step 2.1.2.3.3

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

$32 - 13 \cdot 8 + 72$

Step 2.1.2.3.4

Multiply $- 13$ by $8$ .

$32 - 104 + 72$

Step 2.1.2.3.5

Subtract $104$ from $32$ .

$- 72 + 72$

Step 2.1.2.3.6

Add $- 72$ and $72$ .

$0$

Step 2.1.2.4

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

$\frac{r^{5} - 13 r^{3} + 72}{r - 2}$

Step 2.1.2.5

Divide $r^{5} - 13 r^{3} + 72$ by $r - 2$ .

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Step 2.1.2.5.1

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

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

Step 2.1.2.5.2

Divide the highest order term in the dividend $r^{5}$ by the highest order term in divisor $r$ .

$r^{4}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

Step 2.1.2.5.3

Multiply the new quotient term by the divisor.

$r^{4}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

Step 2.1.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $r^{5} - 2 r^{4}$

$r^{4}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

Step 2.1.2.5.5

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

$r^{4}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

Step 2.1.2.5.6

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

$r^{4}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

Step 2.1.2.5.7

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

$r^{4}$

$2 r^{3}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

Step 2.1.2.5.8

Multiply the new quotient term by the divisor.

$r^{4}$

$2 r^{3}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

Step 2.1.2.5.9

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

$r^{4}$

$2 r^{3}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

Step 2.1.2.5.10

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

$r^{4}$

$2 r^{3}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

Step 2.1.2.5.11

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

$r^{4}$

$2 r^{3}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

Step 2.1.2.5.12

Divide the highest order term in the dividend $- 9 r^{3}$ by the highest order term in divisor $r$ .

$r^{4}$

$2 r^{3}$

$9 r^{2}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

Step 2.1.2.5.13

Multiply the new quotient term by the divisor.

$r^{4}$

$2 r^{3}$

$9 r^{2}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

$9 r^{3}$

$18 r^{2}$

Step 2.1.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 r^{3} + 18 r^{2}$

$r^{4}$

$2 r^{3}$

$9 r^{2}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

$9 r^{3}$

$18 r^{2}$

Step 2.1.2.5.15

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

$r^{4}$

$2 r^{3}$

$9 r^{2}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

$9 r^{3}$

$18 r^{2}$

Step 2.1.2.5.16

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

$r^{4}$

$2 r^{3}$

$9 r^{2}$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

$9 r^{3}$

$18 r^{2}$

$0 r$

Step 2.1.2.5.17

Divide the highest order term in the dividend $- 18 r^{2}$ by the highest order term in divisor $r$ .

$r^{4}$

$2 r^{3}$

$9 r^{2}$

$18 r$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

$9 r^{3}$

$18 r^{2}$

$0 r$

Step 2.1.2.5.18

Multiply the new quotient term by the divisor.

$r^{4}$

$2 r^{3}$

$9 r^{2}$

$18 r$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

$9 r^{3}$

$18 r^{2}$

$0 r$

$18 r^{2}$

$36 r$

Step 2.1.2.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 r^{2} + 36 r$

$r^{4}$

$2 r^{3}$

$9 r^{2}$

$18 r$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

$9 r^{3}$

$18 r^{2}$

$0 r$

$18 r^{2}$

$36 r$

Step 2.1.2.5.20

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

$r^{4}$

$2 r^{3}$

$9 r^{2}$

$18 r$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

$9 r^{3}$

$18 r^{2}$

$0 r$

$18 r^{2}$

$36 r$

Step 2.1.2.5.21

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

$r^{4}$

$2 r^{3}$

$9 r^{2}$

$18 r$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

$9 r^{3}$

$18 r^{2}$

$0 r$

$18 r^{2}$

$36 r$

$72$

Step 2.1.2.5.22

Divide the highest order term in the dividend $- 36 r$ by the highest order term in divisor $r$ .

$r^{4}$

$2 r^{3}$

$9 r^{2}$

$18 r$

$36$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

$9 r^{3}$

$18 r^{2}$

$0 r$

$18 r^{2}$

$36 r$

$72$

Step 2.1.2.5.23

Multiply the new quotient term by the divisor.

$r^{4}$

$2 r^{3}$

$9 r^{2}$

$18 r$

$36$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

$9 r^{3}$

$18 r^{2}$

$0 r$

$18 r^{2}$

$36 r$

$72$

$36 r$

$72$

Step 2.1.2.5.24

The expression needs to be subtracted from the dividend, so change all the signs in $- 36 r + 72$

$r^{4}$

$2 r^{3}$

$9 r^{2}$

$18 r$

$36$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

$9 r^{3}$

$18 r^{2}$

$0 r$

$18 r^{2}$

$36 r$

$72$

$36 r$

$72$

Step 2.1.2.5.25

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

$r^{4}$

$2 r^{3}$

$9 r^{2}$

$18 r$

$36$

$r$

$2$

$r^{5}$

$0 r^{4}$

$13 r^{3}$

$0 r^{2}$

$0 r$

$72$

$r^{5}$

$2 r^{4}$

$13 r^{3}$

$2 r^{4}$

$4 r^{3}$

$9 r^{3}$

$0 r^{2}$

$9 r^{3}$

$18 r^{2}$

$0 r$

$18 r^{2}$

$36 r$

$72$

$36 r$

$72$

$0$

Step 2.1.2.5.26

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

$r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36$

Step 2.1.2.6

Write $r^{5} - 13 r^{3} + 72$ as a set of factors.

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36) + 4 r^{4} - 10 r^{2} - 12 r = 0$

Step 2.1.3

Factor $2 r$ out of $4 r^{4} - 10 r^{2} - 12 r$ .

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Step 2.1.3.1

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

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36) + 2 r (2 r^{3}) - 10 r^{2} - 12 r = 0$

Step 2.1.3.2

Factor $2 r$ out of $- 10 r^{2}$ .

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36) + 2 r (2 r^{3}) + 2 r (- 5 r) - 12 r = 0$

Step 2.1.3.3

Factor $2 r$ out of $- 12 r$ .

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36) + 2 r (2 r^{3}) + 2 r (- 5 r) + 2 r (- 6) = 0$

Step 2.1.3.4

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

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36) + 2 r (2 r^{3} - 5 r) + 2 r (- 6) = 0$

Step 2.1.3.5

Factor $2 r$ out of $2 r (2 r^{3} - 5 r) + 2 r (- 6)$ .

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36) + 2 r (2 r^{3} - 5 r - 6) = 0$

Step 2.1.4

Factor.

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Step 2.1.4.1

Factor $2 r^{3} - 5 r - 6$ using the rational roots test.

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Step 2.1.4.1.1

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.

$p = \pm 1, \pm 6, \pm 2, \pm 3$

$q = \pm 1, \pm 2$

Step 2.1.4.1.2

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

$\pm 1, \pm 0.5, \pm 6, \pm 3, \pm 2, \pm 1.5$

Step 2.1.4.1.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 2.1.4.1.3.1

Substitute $2$ into the polynomial.

$2 \cdot 2^{3} - 5 \cdot 2 - 6$

Step 2.1.4.1.3.2

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

$2 \cdot 8 - 5 \cdot 2 - 6$

Step 2.1.4.1.3.3

Multiply $2$ by $8$ .

$16 - 5 \cdot 2 - 6$

Step 2.1.4.1.3.4

Multiply $- 5$ by $2$ .

$16 - 10 - 6$

Step 2.1.4.1.3.5

Subtract $10$ from $16$ .

$6 - 6$

Step 2.1.4.1.3.6

Subtract $6$ from $6$ .

$0$

Step 2.1.4.1.4

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

$\frac{2 r^{3} - 5 r - 6}{r - 2}$

Step 2.1.4.1.5

Divide $2 r^{3} - 5 r - 6$ by $r - 2$ .

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Step 2.1.4.1.5.1

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

$r$

$2$

$2 r^{3}$

$0 r^{2}$

$5 r$

$6$

Step 2.1.4.1.5.2

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

					$2 r^{2}$
	$r$	-	$2$		$2 r^{3}$	+	$0 r^{2}$	-	$5 r$	-	$6$

Step 2.1.4.1.5.3

Multiply the new quotient term by the divisor.

				$2 r^{2}$
$r$	-	$2$		$2 r^{3}$	+	$0 r^{2}$	-	$5 r$	-	$6$
			+	$2 r^{3}$	-	$4 r^{2}$

Step 2.1.4.1.5.4

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

				$2 r^{2}$
$r$	-	$2$		$2 r^{3}$	+	$0 r^{2}$	-	$5 r$	-	$6$
			-	$2 r^{3}$	+	$4 r^{2}$

Step 2.1.4.1.5.5

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

				$2 r^{2}$
$r$	-	$2$		$2 r^{3}$	+	$0 r^{2}$	-	$5 r$	-	$6$
			-	$2 r^{3}$	+	$4 r^{2}$
					+	$4 r^{2}$

Step 2.1.4.1.5.6

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

				$2 r^{2}$
$r$	-	$2$		$2 r^{3}$	+	$0 r^{2}$	-	$5 r$	-	$6$
			-	$2 r^{3}$	+	$4 r^{2}$
					+	$4 r^{2}$	-	$5 r$

Step 2.1.4.1.5.7

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

				$2 r^{2}$	+	$4 r$
$r$	-	$2$		$2 r^{3}$	+	$0 r^{2}$	-	$5 r$	-	$6$
			-	$2 r^{3}$	+	$4 r^{2}$
					+	$4 r^{2}$	-	$5 r$

Step 2.1.4.1.5.8

Multiply the new quotient term by the divisor.

				$2 r^{2}$	+	$4 r$
$r$	-	$2$		$2 r^{3}$	+	$0 r^{2}$	-	$5 r$	-	$6$
			-	$2 r^{3}$	+	$4 r^{2}$
					+	$4 r^{2}$	-	$5 r$
					+	$4 r^{2}$	-	$8 r$

Step 2.1.4.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $4 r^{2} - 8 r$

				$2 r^{2}$	+	$4 r$
$r$	-	$2$		$2 r^{3}$	+	$0 r^{2}$	-	$5 r$	-	$6$
			-	$2 r^{3}$	+	$4 r^{2}$
					+	$4 r^{2}$	-	$5 r$
					-	$4 r^{2}$	+	$8 r$

Step 2.1.4.1.5.10

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

				$2 r^{2}$	+	$4 r$
$r$	-	$2$		$2 r^{3}$	+	$0 r^{2}$	-	$5 r$	-	$6$
			-	$2 r^{3}$	+	$4 r^{2}$
					+	$4 r^{2}$	-	$5 r$
					-	$4 r^{2}$	+	$8 r$
							+	$3 r$

Step 2.1.4.1.5.11

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

				$2 r^{2}$	+	$4 r$
$r$	-	$2$		$2 r^{3}$	+	$0 r^{2}$	-	$5 r$	-	$6$
			-	$2 r^{3}$	+	$4 r^{2}$
					+	$4 r^{2}$	-	$5 r$
					-	$4 r^{2}$	+	$8 r$
							+	$3 r$	-	$6$

Step 2.1.4.1.5.12

Divide the highest order term in the dividend $3 r$ by the highest order term in divisor $r$ .

				$2 r^{2}$	+	$4 r$	+	$3$
$r$	-	$2$		$2 r^{3}$	+	$0 r^{2}$	-	$5 r$	-	$6$
			-	$2 r^{3}$	+	$4 r^{2}$
					+	$4 r^{2}$	-	$5 r$
					-	$4 r^{2}$	+	$8 r$
							+	$3 r$	-	$6$

Step 2.1.4.1.5.13

Multiply the new quotient term by the divisor.

				$2 r^{2}$	+	$4 r$	+	$3$
$r$	-	$2$		$2 r^{3}$	+	$0 r^{2}$	-	$5 r$	-	$6$
			-	$2 r^{3}$	+	$4 r^{2}$
					+	$4 r^{2}$	-	$5 r$
					-	$4 r^{2}$	+	$8 r$
							+	$3 r$	-	$6$
							+	$3 r$	-	$6$

Step 2.1.4.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $3 r - 6$

				$2 r^{2}$	+	$4 r$	+	$3$
$r$	-	$2$		$2 r^{3}$	+	$0 r^{2}$	-	$5 r$	-	$6$
			-	$2 r^{3}$	+	$4 r^{2}$
					+	$4 r^{2}$	-	$5 r$
					-	$4 r^{2}$	+	$8 r$
							+	$3 r$	-	$6$
							-	$3 r$	+	$6$

Step 2.1.4.1.5.15

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

				$2 r^{2}$	+	$4 r$	+	$3$
$r$	-	$2$		$2 r^{3}$	+	$0 r^{2}$	-	$5 r$	-	$6$
			-	$2 r^{3}$	+	$4 r^{2}$
					+	$4 r^{2}$	-	$5 r$
					-	$4 r^{2}$	+	$8 r$
							+	$3 r$	-	$6$
							-	$3 r$	+	$6$
										$0$

Step 2.1.4.1.5.16

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

$2 r^{2} + 4 r + 3$

Step 2.1.4.1.6

Write $2 r^{3} - 5 r - 6$ as a set of factors.

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36) + 2 r ((r - 2) (2 r^{2} + 4 r + 3)) = 0$

Step 2.1.4.2

Remove unnecessary parentheses.

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36) + 2 r (r - 2) (2 r^{2} + 4 r + 3) = 0$

Step 2.1.5

Factor $r - 2$ out of $(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36) + 2 r (r - 2) (2 r^{2} + 4 r + 3)$ .

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Step 2.1.5.1

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

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36) + (r - 2) (2 r (2 r^{2} + 4 r + 3)) = 0$

Step 2.1.5.2

Factor $r - 2$ out of $(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36) + (r - 2) (2 r (2 r^{2} + 4 r + 3))$ .

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36 + 2 r (2 r^{2} + 4 r + 3)) = 0$

Step 2.1.6

Apply the distributive property.

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36 + 2 r (2 r^{2}) + 2 r (4 r) + 2 r \cdot 3) = 0$

Step 2.1.7

Simplify.

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Step 2.1.7.1

Rewrite using the commutative property of multiplication.

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36 + 2 \cdot (2 r \cdot r^{2}) + 2 r (4 r) + 2 r \cdot 3) = 0$

Step 2.1.7.2

Rewrite using the commutative property of multiplication.

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36 + 2 \cdot (2 r \cdot r^{2}) + 2 \cdot (4 r \cdot r) + 2 r \cdot 3) = 0$

Step 2.1.7.3

Multiply $3$ by $2$ .

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36 + 2 \cdot (2 r \cdot r^{2}) + 2 \cdot (4 r \cdot r) + 6 r) = 0$

Step 2.1.8

Simplify each term.

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Step 2.1.8.1

Multiply $r$ by $r^{2}$ by adding the exponents.

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Step 2.1.8.1.1

Move $r^{2}$ .

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36 + 2 \cdot (2 (r^{2} r)) + 2 \cdot (4 r \cdot r) + 6 r) = 0$

Step 2.1.8.1.2

Multiply $r^{2}$ by $r$ .

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Step 2.1.8.1.2.1

Raise $r$ to the power of $1$ .

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36 + 2 \cdot (2 (r^{2} r)) + 2 \cdot (4 r \cdot r) + 6 r) = 0$

Step 2.1.8.1.2.2

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

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36 + 2 \cdot (2 r^{2 + 1}) + 2 \cdot (4 r \cdot r) + 6 r) = 0$

Step 2.1.8.1.3

Add $2$ and $1$ .

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36 + 2 \cdot (2 r^{3}) + 2 \cdot (4 r \cdot r) + 6 r) = 0$

Step 2.1.8.2

Multiply $2$ by $2$ .

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36 + 4 r^{3} + 2 \cdot (4 r \cdot r) + 6 r) = 0$

Step 2.1.8.3

Multiply $r$ by $r$ by adding the exponents.

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Step 2.1.8.3.1

Move $r$ .

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36 + 4 r^{3} + 2 \cdot (4 (r \cdot r)) + 6 r) = 0$

Step 2.1.8.3.2

Multiply $r$ by $r$ .

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36 + 4 r^{3} + 2 \cdot (4 r^{2}) + 6 r) = 0$

Step 2.1.8.4

Multiply $2$ by $4$ .

$(r - 2) (r^{4} + 2 r^{3} - 9 r^{2} - 18 r - 36 + 4 r^{3} + 8 r^{2} + 6 r) = 0$

Step 2.1.9

Add $2 r^{3}$ and $4 r^{3}$ .

$(r - 2) (r^{4} + 6 r^{3} - 9 r^{2} - 18 r - 36 + 8 r^{2} + 6 r) = 0$

Step 2.1.10

Add $- 9 r^{2}$ and $8 r^{2}$ .

$(r - 2) (r^{4} + 6 r^{3} - r^{2} - 18 r - 36 + 6 r) = 0$

Step 2.1.11

Add $- 18 r$ and $6 r$ .

$(r - 2) (r^{4} + 6 r^{3} - r^{2} - 12 r - 36) = 0$

Step 2.2

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

$r - 2 = 0$

$r^{4} + 6 r^{3} - r^{2} - 12 r - 36 = 0$

Step 2.3

Set $r - 2$ equal to $0$ and solve for $r$ .

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Step 2.3.1

Set $r - 2$ equal to $0$ .

$r - 2 = 0$

Step 2.3.2

Add $2$ to both sides of the equation.

$r = 2$

Step 2.4

Set $r^{4} + 6 r^{3} - r^{2} - 12 r - 36$ equal to $0$ and solve for $r$ .

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Step 2.4.1

Set $r^{4} + 6 r^{3} - r^{2} - 12 r - 36$ equal to $0$ .

$r^{4} + 6 r^{3} - r^{2} - 12 r - 36 = 0$

Step 2.4.2

Solve $r^{4} + 6 r^{3} - r^{2} - 12 r - 36 = 0$ for $r$ .

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Step 2.4.2.1

Factor the left side of the equation.

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Step 2.4.2.1.1

Regroup terms.

$r^{4} + 6 r^{3} - r^{2} - 12 r - 36 = 0$

Step 2.4.2.1.2

Factor $r^{3}$ out of $r^{4} + 6 r^{3}$ .

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Step 2.4.2.1.2.1

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

$r^{3} r + 6 r^{3} - r^{2} - 12 r - 36 = 0$

Step 2.4.2.1.2.2

Factor $r^{3}$ out of $6 r^{3}$ .

$r^{3} r + r^{3} \cdot 6 - r^{2} - 12 r - 36 = 0$

Step 2.4.2.1.2.3

Factor $r^{3}$ out of $r^{3} r + r^{3} \cdot 6$ .

$r^{3} (r + 6) - r^{2} - 12 r - 36 = 0$

Step 2.4.2.1.3

Factor by grouping.

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Step 2.4.2.1.3.1

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 $a \cdot c = - 1 \cdot - 36 = 36$ and whose sum is $b = - 12$ .

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Step 2.4.2.1.3.1.1

Factor $- 12$ out of $- 12 r$ .

$r^{3} (r + 6) - r^{2} - 12 r - 36 = 0$

Step 2.4.2.1.3.1.2

Rewrite $- 12$ as $- 6$ plus $- 6$

$r^{3} (r + 6) - r^{2} + (- 6 - 6) r - 36 = 0$

Step 2.4.2.1.3.1.3

Apply the distributive property.

$r^{3} (r + 6) - r^{2} - 6 r - 6 r - 36 = 0$

Step 2.4.2.1.3.2

Factor out the greatest common factor from each group.

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Step 2.4.2.1.3.2.1

Group the first two terms and the last two terms.

$r^{3} (r + 6) - r^{2} - 6 r - 6 r - 36 = 0$

Step 2.4.2.1.3.2.2

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

$r^{3} (r + 6) + r (- r - 6) + 6 (- r - 6) = 0$

Step 2.4.2.1.3.3

Factor the polynomial by factoring out the greatest common factor, $- r - 6$ .

$r^{3} (r + 6) + (- r - 6) (r + 6) = 0$

Step 2.4.2.1.4

Factor $r + 6$ out of $r^{3} (r + 6) + (- r - 6) (r + 6)$ .

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Step 2.4.2.1.4.1

Factor $r + 6$ out of $r^{3} (r + 6)$ .

$(r + 6) r^{3} + (- r - 6) (r + 6) = 0$

Step 2.4.2.1.4.2

Factor $r + 6$ out of $(- r - 6) (r + 6)$ .

$(r + 6) r^{3} + (r + 6) (- r - 6) = 0$

Step 2.4.2.1.4.3

Factor $r + 6$ out of $(r + 6) r^{3} + (r + 6) (- r - 6)$ .

$(r + 6) (r^{3} - r - 6) = 0$

Step 2.4.2.1.5

Factor.

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Step 2.4.2.1.5.1

Factor $r^{3} - r - 6$ using the rational roots test.

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Step 2.4.2.1.5.1.1

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.

$p = \pm 1, \pm 6, \pm 2, \pm 3$

$q = \pm 1$

Step 2.4.2.1.5.1.2

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

$\pm 1, \pm 6, \pm 2, \pm 3$

Step 2.4.2.1.5.1.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 2.4.2.1.5.1.3.1

Substitute $2$ into the polynomial.

$2^{3} - 1 \cdot 2 - 6$

Step 2.4.2.1.5.1.3.2

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

$8 - 1 \cdot 2 - 6$

Step 2.4.2.1.5.1.3.3

Multiply $- 1$ by $2$ .

$8 - 2 - 6$

Step 2.4.2.1.5.1.3.4

Subtract $2$ from $8$ .

$6 - 6$

Step 2.4.2.1.5.1.3.5

Subtract $6$ from $6$ .

$0$

Step 2.4.2.1.5.1.4

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

$\frac{r^{3} - r - 6}{r - 2}$

Step 2.4.2.1.5.1.5

Divide $r^{3} - r - 6$ by $r - 2$ .

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Step 2.4.2.1.5.1.5.1

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

$r$

$2$

$r^{3}$

$0 r^{2}$

$r$

$6$

Step 2.4.2.1.5.1.5.2

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

					$r^{2}$
	$r$	-	$2$		$r^{3}$	+	$0 r^{2}$	-	$r$	-	$6$

Step 2.4.2.1.5.1.5.3

Multiply the new quotient term by the divisor.

				$r^{2}$
$r$	-	$2$		$r^{3}$	+	$0 r^{2}$	-	$r$	-	$6$
			+	$r^{3}$	-	$2 r^{2}$

Step 2.4.2.1.5.1.5.4

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

				$r^{2}$
$r$	-	$2$		$r^{3}$	+	$0 r^{2}$	-	$r$	-	$6$
			-	$r^{3}$	+	$2 r^{2}$

Step 2.4.2.1.5.1.5.5

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

				$r^{2}$
$r$	-	$2$		$r^{3}$	+	$0 r^{2}$	-	$r$	-	$6$
			-	$r^{3}$	+	$2 r^{2}$
					+	$2 r^{2}$

Step 2.4.2.1.5.1.5.6

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

				$r^{2}$
$r$	-	$2$		$r^{3}$	+	$0 r^{2}$	-	$r$	-	$6$
			-	$r^{3}$	+	$2 r^{2}$
					+	$2 r^{2}$	-	$r$

Step 2.4.2.1.5.1.5.7

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

				$r^{2}$	+	$2 r$
$r$	-	$2$		$r^{3}$	+	$0 r^{2}$	-	$r$	-	$6$
			-	$r^{3}$	+	$2 r^{2}$
					+	$2 r^{2}$	-	$r$

Step 2.4.2.1.5.1.5.8

Multiply the new quotient term by the divisor.

				$r^{2}$	+	$2 r$
$r$	-	$2$		$r^{3}$	+	$0 r^{2}$	-	$r$	-	$6$
			-	$r^{3}$	+	$2 r^{2}$
					+	$2 r^{2}$	-	$r$
					+	$2 r^{2}$	-	$4 r$

Step 2.4.2.1.5.1.5.9

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

				$r^{2}$	+	$2 r$
$r$	-	$2$		$r^{3}$	+	$0 r^{2}$	-	$r$	-	$6$
			-	$r^{3}$	+	$2 r^{2}$
					+	$2 r^{2}$	-	$r$
					-	$2 r^{2}$	+	$4 r$

Step 2.4.2.1.5.1.5.10

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

				$r^{2}$	+	$2 r$
$r$	-	$2$		$r^{3}$	+	$0 r^{2}$	-	$r$	-	$6$
			-	$r^{3}$	+	$2 r^{2}$
					+	$2 r^{2}$	-	$r$
					-	$2 r^{2}$	+	$4 r$
							+	$3 r$

Step 2.4.2.1.5.1.5.11

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

				$r^{2}$	+	$2 r$
$r$	-	$2$		$r^{3}$	+	$0 r^{2}$	-	$r$	-	$6$
			-	$r^{3}$	+	$2 r^{2}$
					+	$2 r^{2}$	-	$r$
					-	$2 r^{2}$	+	$4 r$
							+	$3 r$	-	$6$

Step 2.4.2.1.5.1.5.12

Divide the highest order term in the dividend $3 r$ by the highest order term in divisor $r$ .

				$r^{2}$	+	$2 r$	+	$3$
$r$	-	$2$		$r^{3}$	+	$0 r^{2}$	-	$r$	-	$6$
			-	$r^{3}$	+	$2 r^{2}$
					+	$2 r^{2}$	-	$r$
					-	$2 r^{2}$	+	$4 r$
							+	$3 r$	-	$6$

Step 2.4.2.1.5.1.5.13

Multiply the new quotient term by the divisor.

				$r^{2}$	+	$2 r$	+	$3$
$r$	-	$2$		$r^{3}$	+	$0 r^{2}$	-	$r$	-	$6$
			-	$r^{3}$	+	$2 r^{2}$
					+	$2 r^{2}$	-	$r$
					-	$2 r^{2}$	+	$4 r$
							+	$3 r$	-	$6$
							+	$3 r$	-	$6$

Step 2.4.2.1.5.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $3 r - 6$

				$r^{2}$	+	$2 r$	+	$3$
$r$	-	$2$		$r^{3}$	+	$0 r^{2}$	-	$r$	-	$6$
			-	$r^{3}$	+	$2 r^{2}$
					+	$2 r^{2}$	-	$r$
					-	$2 r^{2}$	+	$4 r$
							+	$3 r$	-	$6$
							-	$3 r$	+	$6$

Step 2.4.2.1.5.1.5.15

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

				$r^{2}$	+	$2 r$	+	$3$
$r$	-	$2$		$r^{3}$	+	$0 r^{2}$	-	$r$	-	$6$
			-	$r^{3}$	+	$2 r^{2}$
					+	$2 r^{2}$	-	$r$
					-	$2 r^{2}$	+	$4 r$
							+	$3 r$	-	$6$
							-	$3 r$	+	$6$
										$0$

Step 2.4.2.1.5.1.5.16

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

$r^{2} + 2 r + 3$

Step 2.4.2.1.5.1.6

Write $r^{3} - r - 6$ as a set of factors.

$(r + 6) ((r - 2) (r^{2} + 2 r + 3)) = 0$

Step 2.4.2.1.5.2

Remove unnecessary parentheses.

$(r + 6) (r - 2) (r^{2} + 2 r + 3) = 0$

Step 2.4.2.2

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

$r + 6 = 0$

$r - 2 = 0$

$r^{2} + 2 r + 3 = 0$

Step 2.4.2.3

Set $r + 6$ equal to $0$ and solve for $r$ .

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Step 2.4.2.3.1

Set $r + 6$ equal to $0$ .

$r + 6 = 0$

Step 2.4.2.3.2

Subtract $6$ from both sides of the equation.

$r = - 6$

Step 2.4.2.4

Set $r - 2$ equal to $0$ and solve for $r$ .

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Step 2.4.2.4.1

Set $r - 2$ equal to $0$ .

$r - 2 = 0$

Step 2.4.2.4.2

Add $2$ to both sides of the equation.

$r = 2$

Step 2.4.2.5

Set $r^{2} + 2 r + 3$ equal to $0$ and solve for $r$ .

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Step 2.4.2.5.1

Set $r^{2} + 2 r + 3$ equal to $0$ .

$r^{2} + 2 r + 3 = 0$

Step 2.4.2.5.2

Solve $r^{2} + 2 r + 3 = 0$ for $r$ .

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Step 2.4.2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.2.5.2.2

Substitute the values $a = 1$ , $b = 2$ , and $c = 3$ into the quadratic formula and solve for $r$ .

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

Step 2.4.2.5.2.3

Simplify.

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Step 2.4.2.5.2.3.1

Simplify the numerator.

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Step 2.4.2.5.2.3.1.1

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

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

Step 2.4.2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 2.4.2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.4.2.5.2.3.1.2.2

Multiply $- 4$ by $3$ .

$r = \frac{- 2 \pm \sqrt{4 - 12}}{2 \cdot 1}$

Step 2.4.2.5.2.3.1.3

Subtract $12$ from $4$ .

$r = \frac{- 2 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 2.4.2.5.2.3.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$r = \frac{- 2 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 2.4.2.5.2.3.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$r = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 2.4.2.5.2.3.1.6

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

$r = \frac{- 2 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 2.4.2.5.2.3.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.4.2.5.2.3.1.7.1

Factor $4$ out of $8$ .

$r = \frac{- 2 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 2.4.2.5.2.3.1.7.2

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

$r = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 2.4.2.5.2.3.1.8

Pull terms out from under the radical.

$r = \frac{- 2 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Step 2.4.2.5.2.3.1.9

Move $2$ to the left of $i$ .

$r = \frac{- 2 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Step 2.4.2.5.2.3.2

Multiply $2$ by $1$ .

$r = \frac{- 2 \pm 2 i \sqrt{2}}{2}$

Step 2.4.2.5.2.3.3

Simplify $\frac{- 2 \pm 2 i \sqrt{2}}{2}$ .

$r = - 1 \pm i \sqrt{2}$

Step 2.4.2.5.2.4

The final answer is the combination of both solutions.

$r = - 1 + i \sqrt{2}, - 1 - i \sqrt{2}$

Step 2.4.2.6

The final solution is all the values that make $(r + 6) (r - 2) (r^{2} + 2 r + 3) = 0$ true.

$r = - 6, 2, - 1 + i \sqrt{2}, - 1 - i \sqrt{2}$

Step 2.5

The final solution is all the values that make $(r - 2) (r^{4} + 6 r^{3} - r^{2} - 12 r - 36) = 0$ true.

$r = 2, - 6, - 1 + i \sqrt{2}, - 1 - i \sqrt{2}$

Step 3