Algebra Examples

$x^{4} - 9 x^{3} + 12 x^{2} + 80 x - 192$

Step 1

Regroup terms.

$12 x^{2} - 192 + x^{4} - 9 x^{3} + 80 x$

Step 2

Factor $12$ out of $12 x^{2} - 192$ .

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Step 2.1

Factor $12$ out of $12 x^{2}$ .

$12 (x^{2}) - 192 + x^{4} - 9 x^{3} + 80 x$

Step 2.2

Factor $12$ out of $- 192$ .

$12 x^{2} + 12 \cdot - 16 + x^{4} - 9 x^{3} + 80 x$

Step 2.3

Factor $12$ out of $12 x^{2} + 12 \cdot - 16$ .

$12 (x^{2} - 16) + x^{4} - 9 x^{3} + 80 x$

Step 3

Rewrite $16$ as $4^{2}$ .

$12 (x^{2} - 4^{2}) + x^{4} - 9 x^{3} + 80 x$

Step 4

Factor.

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Step 4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$12 ((x + 4) (x - 4)) + x^{4} - 9 x^{3} + 80 x$

Step 4.2

Remove unnecessary parentheses.

$12 (x + 4) (x - 4) + x^{4} - 9 x^{3} + 80 x$

Step 5

Factor $x$ out of $x^{4} - 9 x^{3} + 80 x$ .

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Step 5.1

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

$12 (x + 4) (x - 4) + x \cdot x^{3} - 9 x^{3} + 80 x$

Step 5.2

Factor $x$ out of $- 9 x^{3}$ .

$12 (x + 4) (x - 4) + x \cdot x^{3} + x (- 9 x^{2}) + 80 x$

Step 5.3

Factor $x$ out of $80 x$ .

$12 (x + 4) (x - 4) + x \cdot x^{3} + x (- 9 x^{2}) + x \cdot 80$

Step 5.4

Factor $x$ out of $x \cdot x^{3} + x (- 9 x^{2})$ .

$12 (x + 4) (x - 4) + x (x^{3} - 9 x^{2}) + x \cdot 80$

Step 5.5

Factor $x$ out of $x (x^{3} - 9 x^{2}) + x \cdot 80$ .

$12 (x + 4) (x - 4) + x (x^{3} - 9 x^{2} + 80)$

Step 6

Factor.

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Step 6.1

Factor $x^{3} - 9 x^{2} + 80$ using the rational roots test.

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Step 6.1.1

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.

$p = \pm 1, \pm 80, \pm 2, \pm 40, \pm 4, \pm 20, \pm 5, \pm 16, \pm 8, \pm 10$

$q = \pm 1$

Step 6.1.2

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

$\pm 1, \pm 80, \pm 2, \pm 40, \pm 4, \pm 20, \pm 5, \pm 16, \pm 8, \pm 10$

Step 6.1.3

Substitute $4$ and simplify the expression. In this case, the expression is equal to $0$ so $4$ is a root of the polynomial.

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Step 6.1.3.1

Substitute $4$ into the polynomial.

$4^{3} - 9 \cdot 4^{2} + 80$

Step 6.1.3.2

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

$64 - 9 \cdot 4^{2} + 80$

Step 6.1.3.3

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

$64 - 9 \cdot 16 + 80$

Step 6.1.3.4

Multiply $- 9$ by $16$ .

$64 - 144 + 80$

Step 6.1.3.5

Subtract $144$ from $64$ .

$- 80 + 80$

Step 6.1.3.6

Add $- 80$ and $80$ .

$0$

Step 6.1.4

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

$\frac{x^{3} - 9 x^{2} + 80}{x - 4}$

Step 6.1.5

Divide $x^{3} - 9 x^{2} + 80$ by $x - 4$ .

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Step 6.1.5.1

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

$x$

$4$

$x^{3}$

$9 x^{2}$

$0 x$

$80$

Step 6.1.5.2

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

					$x^{2}$
	$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$80$

Step 6.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$80$
			+	$x^{3}$	-	$4 x^{2}$

Step 6.1.5.4

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$80$
			-	$x^{3}$	+	$4 x^{2}$

Step 6.1.5.5

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$80$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$

Step 6.1.5.6

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$80$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$0 x$

Step 6.1.5.7

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$5 x$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$80$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$0 x$

Step 6.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$80$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					-	$5 x^{2}$	+	$20 x$

Step 6.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{2} + 20 x$

				$x^{2}$	-	$5 x$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$80$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	-	$20 x$

Step 6.1.5.10

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

				$x^{2}$	-	$5 x$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$80$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	-	$20 x$
							-	$20 x$

Step 6.1.5.11

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

				$x^{2}$	-	$5 x$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$80$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	-	$20 x$
							-	$20 x$	+	$80$

Step 6.1.5.12

Divide the highest order term in the dividend $- 20 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$5 x$	-	$20$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$80$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	-	$20 x$
							-	$20 x$	+	$80$

Step 6.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$	-	$20$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$80$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	-	$20 x$
							-	$20 x$	+	$80$
							-	$20 x$	+	$80$

Step 6.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 x + 80$

				$x^{2}$	-	$5 x$	-	$20$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$80$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	-	$20 x$
							-	$20 x$	+	$80$
							+	$20 x$	-	$80$

Step 6.1.5.15

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

				$x^{2}$	-	$5 x$	-	$20$
$x$	-	$4$		$x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$80$
			-	$x^{3}$	+	$4 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	-	$20 x$
							-	$20 x$	+	$80$
							+	$20 x$	-	$80$
										$0$

Step 6.1.5.16

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

$x^{2} - 5 x - 20$

Step 6.1.6

Write $x^{3} - 9 x^{2} + 80$ as a set of factors.

$12 (x + 4) (x - 4) + x ((x - 4) (x^{2} - 5 x - 20))$

Step 6.2

Remove unnecessary parentheses.

$12 (x + 4) (x - 4) + x (x - 4) (x^{2} - 5 x - 20)$

Step 7

Factor $x - 4$ out of $12 (x + 4) (x - 4) + x (x - 4) (x^{2} - 5 x - 20)$ .

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Step 7.1

Factor $x - 4$ out of $12 (x + 4) (x - 4)$ .

$(x - 4) (12 (x + 4)) + x (x - 4) (x^{2} - 5 x - 20)$

Step 7.2

Factor $x - 4$ out of $x (x - 4) (x^{2} - 5 x - 20)$ .

$(x - 4) (12 (x + 4)) + (x - 4) (x (x^{2} - 5 x - 20))$

Step 7.3

Factor $x - 4$ out of $(x - 4) (12 (x + 4)) + (x - 4) (x (x^{2} - 5 x - 20))$ .

$(x - 4) (12 (x + 4) + x (x^{2} - 5 x - 20))$

Step 8

Apply the distributive property.

$(x - 4) (12 x + 12 \cdot 4 + x (x^{2} - 5 x - 20))$

Step 9

Multiply $12$ by $4$ .

$(x - 4) (12 x + 48 + x (x^{2} - 5 x - 20))$

Step 10

Apply the distributive property.

$(x - 4) (12 x + 48 + x \cdot x^{2} + x (- 5 x) + x \cdot - 20)$

Step 11

Simplify.

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Step 11.1

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

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Step 11.1.1

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

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Step 11.1.1.1

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

$(x - 4) (12 x + 48 + x^{1} x^{2} + x (- 5 x) + x \cdot - 20)$

Step 11.1.1.2

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

$(x - 4) (12 x + 48 + x^{1 + 2} + x (- 5 x) + x \cdot - 20)$

Step 11.1.2

Add $1$ and $2$ .

$(x - 4) (12 x + 48 + x^{3} + x (- 5 x) + x \cdot - 20)$

Step 11.2

Rewrite using the commutative property of multiplication.

$(x - 4) (12 x + 48 + x^{3} - 5 x \cdot x + x \cdot - 20)$

Step 11.3

Move $- 20$ to the left of $x$ .

$(x - 4) (12 x + 48 + x^{3} - 5 x \cdot x - 20 \cdot x)$

Step 12

Multiply $x$ by $x$ by adding the exponents.

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Step 12.1

Move $x$ .

$(x - 4) (12 x + 48 + x^{3} - 5 (x \cdot x) - 20 \cdot x)$

Step 12.2

Multiply $x$ by $x$ .

$(x - 4) (12 x + 48 + x^{3} - 5 x^{2} - 20 \cdot x)$

$(x - 4) (12 x + 48 + x^{3} - 5 x^{2} - 20 x)$

Step 13

Subtract $20 x$ from $12 x$ .

$(x - 4) (48 + x^{3} - 5 x^{2} - 8 x)$

Step 14

Reorder terms.

$(x - 4) (x^{3} - 5 x^{2} - 8 x + 48)$

Step 15

Factor.

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Step 15.1

Rewrite $x^{3} - 5 x^{2} - 8 x + 48$ in a factored form.

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Step 15.1.1

Factor $x^{3} - 5 x^{2} - 8 x + 48$ using the rational roots test.

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Step 15.1.1.1

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.

$p = \pm 1, \pm 48, \pm 2, \pm 24, \pm 3, \pm 16, \pm 4, \pm 12, \pm 6, \pm 8$

$q = \pm 1$

Step 15.1.1.2

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

$\pm 1, \pm 48, \pm 2, \pm 24, \pm 3, \pm 16, \pm 4, \pm 12, \pm 6, \pm 8$

Step 15.1.1.3

Substitute $- 3$ and simplify the expression. In this case, the expression is equal to $0$ so $- 3$ is a root of the polynomial.

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Step 15.1.1.3.1

Substitute $- 3$ into the polynomial.

${(- 3)}^{3} - 5 {(- 3)}^{2} - 8 \cdot - 3 + 48$

Step 15.1.1.3.2

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

$- 27 - 5 {(- 3)}^{2} - 8 \cdot - 3 + 48$

Step 15.1.1.3.3

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

$- 27 - 5 \cdot 9 - 8 \cdot - 3 + 48$

Step 15.1.1.3.4

Multiply $- 5$ by $9$ .

$- 27 - 45 - 8 \cdot - 3 + 48$

Step 15.1.1.3.5

Subtract $45$ from $- 27$ .

$- 72 - 8 \cdot - 3 + 48$

Step 15.1.1.3.6

Multiply $- 8$ by $- 3$ .

$- 72 + 24 + 48$

Step 15.1.1.3.7

Add $- 72$ and $24$ .

$- 48 + 48$

Step 15.1.1.3.8

Add $- 48$ and $48$ .

$0$

Step 15.1.1.4

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

$\frac{x^{3} - 5 x^{2} - 8 x + 48}{x + 3}$

Step 15.1.1.5

Divide $x^{3} - 5 x^{2} - 8 x + 48$ by $x + 3$ .

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Step 15.1.1.5.1

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

$x$

$3$

$x^{3}$

$5 x^{2}$

$8 x$

$48$

Step 15.1.1.5.2

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

					$x^{2}$
	$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$

Step 15.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			+	$x^{3}$	+	$3 x^{2}$

Step 15.1.1.5.4

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$

Step 15.1.1.5.5

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$

Step 15.1.1.5.6

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$

Step 15.1.1.5.7

Divide the highest order term in the dividend $- 8 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$8 x$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$

Step 15.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					-	$8 x^{2}$	-	$24 x$

Step 15.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} - 24 x$

				$x^{2}$	-	$8 x$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	+	$24 x$

Step 15.1.1.5.10

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

				$x^{2}$	-	$8 x$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	+	$24 x$
							+	$16 x$

Step 15.1.1.5.11

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

				$x^{2}$	-	$8 x$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	+	$24 x$
							+	$16 x$	+	$48$

Step 15.1.1.5.12

Divide the highest order term in the dividend $16 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	+	$24 x$
							+	$16 x$	+	$48$

Step 15.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	+	$24 x$
							+	$16 x$	+	$48$
							+	$16 x$	+	$48$

Step 15.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $16 x + 48$

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	+	$24 x$
							+	$16 x$	+	$48$
							-	$16 x$	-	$48$

Step 15.1.1.5.15

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

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	+	$24 x$
							+	$16 x$	+	$48$
							-	$16 x$	-	$48$
										$0$

Step 15.1.1.5.16

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

$x^{2} - 8 x + 16$

Step 15.1.1.6

Write $x^{3} - 5 x^{2} - 8 x + 48$ as a set of factors.

$(x - 4) ((x + 3) (x^{2} - 8 x + 16))$

Step 15.1.2

Factor using the perfect square rule.

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Step 15.1.2.1

Rewrite $16$ as $4^{2}$ .

$(x - 4) ((x + 3) (x^{2} - 8 x + 4^{2}))$

Step 15.1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Step 15.1.2.3

Rewrite the polynomial.

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

Step 15.1.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

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

Step 15.2

Remove unnecessary parentheses.

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

Step 16

Multiply $x - 4$ by ${(x - 4)}^{2}$ by adding the exponents.

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Step 16.1

Move ${(x - 4)}^{2}$ .

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

Step 16.2

Multiply ${(x - 4)}^{2}$ by $x - 4$ .

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Step 16.2.1

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

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

Step 16.2.2

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

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

Step 16.3

Add $2$ and $1$ .

${(x - 4)}^{3} (x + 3)$