Write as a Set of Linear Factors x^5-3x^4+4x^3-32x+48

$x^{5} - 3 x^{4} + 4 x^{3} - 32 x + 48$

Step 1

Regroup terms.

$- 3 x^{4} + 48 + x^{5} + 4 x^{3} - 32 x$

Step 2

Factor $- 3$ out of $- 3 x^{4} + 48$ .

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

$- 3 (x^{4}) + 48 + x^{5} + 4 x^{3} - 32 x$

Factor $- 3$ out of $48$ .

$- 3 (x^{4}) - 3 (- 16) + x^{5} + 4 x^{3} - 32 x$

Factor $- 3$ out of $- 3 (x^{4}) - 3 (- 16)$ .

$- 3 (x^{4} - 16) + x^{5} + 4 x^{3} - 32 x$

Step 3

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

$- 3 ({(x^{2})}^{2} - 16) + x^{5} + 4 x^{3} - 32 x$

Step 4

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

$- 3 ({(x^{2})}^{2} - 4^{2}) + x^{5} + 4 x^{3} - 32 x$

Step 5

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

$- 3 ((x^{2} + 4) (x^{2} - 4)) + x^{5} + 4 x^{3} - 32 x$

Step 6

Factor.

Simplify.

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

$- 3 ((x^{2} + 4) (x^{2} - 2^{2})) + x^{5} + 4 x^{3} - 32 x$

Factor.

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

$- 3 ((x^{2} + 4) ((x + 2) (x - 2))) + x^{5} + 4 x^{3} - 32 x$

Remove unnecessary parentheses.

$- 3 ((x^{2} + 4) (x + 2) (x - 2)) + x^{5} + 4 x^{3} - 32 x$

Remove unnecessary parentheses.

$- 3 (x^{2} + 4) (x + 2) (x - 2) + x^{5} + 4 x^{3} - 32 x$

Step 7

Factor $x$ out of $x^{5} + 4 x^{3} - 32 x$ .

Factor $x$ out of $x^{5}$ .

$- 3 (x^{2} + 4) (x + 2) (x - 2) + x \cdot x^{4} + 4 x^{3} - 32 x$

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

$- 3 (x^{2} + 4) (x + 2) (x - 2) + x \cdot x^{4} + x (4 x^{2}) - 32 x$

Factor $x$ out of $- 32 x$ .

$- 3 (x^{2} + 4) (x + 2) (x - 2) + x \cdot x^{4} + x (4 x^{2}) + x \cdot - 32$

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

$- 3 (x^{2} + 4) (x + 2) (x - 2) + x (x^{4} + 4 x^{2}) + x \cdot - 32$

Factor $x$ out of $x (x^{4} + 4 x^{2}) + x \cdot - 32$ .

$- 3 (x^{2} + 4) (x + 2) (x - 2) + x (x^{4} + 4 x^{2} - 32)$

Step 8

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

$- 3 (x^{2} + 4) (x + 2) (x - 2) + x ({(x^{2})}^{2} + 4 x^{2} - 32)$

Step 9

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

$- 3 (x^{2} + 4) (x + 2) (x - 2) + x (u^{2} + 4 u - 32)$

Step 10

Factor $u^{2} + 4 u - 32$ using the AC method.

Consider the form $x^{2} + i x + c$ . Find a pair of integers whose product is $c$ and whose sum is $i$ . In this case, whose product is $- 32$ and whose sum is $4$ .

$- 4, 8$

Write the factored form using these integers.

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

Step 11

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

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

Step 12

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

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

Step 13

Factor.

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

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

Remove unnecessary parentheses.

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

Step 14

Factor $(x + 2) (x - 2)$ out of $- 3 (x^{2} + 4) (x + 2) (x - 2) + x (x + 2) (x - 2) (x^{2} + 8)$ .

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

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

Factor $(x + 2) (x - 2)$ out of $x (x + 2) (x - 2) (x^{2} + 8)$ .

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

Factor $(x + 2) (x - 2)$ out of $(x + 2) (x - 2) (- 3 (x^{2} + 4)) + (x + 2) (x - 2) ((x) (x^{2} + 8))$ .

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

Step 15

Apply the distributive property.

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

Step 16

Multiply $- 3$ by $4$ .

$(x + 2) (x - 2) (- 3 x^{2} - 12 + (x) (x^{2} + 8))$

Step 17

Apply the distributive property.

$(x + 2) (x - 2) (- 3 x^{2} - 12 + x \cdot x^{2} + x \cdot 8)$

Step 18

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

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

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

$(x + 2) (x - 2) (- 3 x^{2} - 12 + x^{1} x^{2} + x \cdot 8)$

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

$(x + 2) (x - 2) (- 3 x^{2} - 12 + x^{1 + 2} + x \cdot 8)$

Add $1$ and $2$ .

$(x + 2) (x - 2) (- 3 x^{2} - 12 + x^{3} + x \cdot 8)$

Step 19

Move $8$ to the left of $x$ .

$(x + 2) (x - 2) (- 3 x^{2} - 12 + x^{3} + 8 x)$

Step 20

Reorder terms.

$(x + 2) (x - 2) (x^{3} - 3 x^{2} + 8 x - 12)$

Step 21

Factor.

Factor $x^{3} - 3 x^{2} + 8 x - 12$ using the rational roots test.

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.

$p = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4 q = \pm 1$

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

$\pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

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

Substitute $2$ into the polynomial.

$2^{3} - 3 \cdot 2^{2} + 8 \cdot 2 - 12$

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

$8 - 3 \cdot 2^{2} + 8 \cdot 2 - 12$

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

$8 - 3 \cdot 4 + 8 \cdot 2 - 12$

Multiply $- 3$ by $4$ .

$8 - 12 + 8 \cdot 2 - 12$

Subtract $12$ from $8$ .

$- 4 + 8 \cdot 2 - 12$

Multiply $8$ by $2$ .

$- 4 + 16 - 12$

Add $- 4$ and $16$ .

$12 - 12$

Subtract $12$ from $12$ .

$0$

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

$\frac{x^{3} - 3 x^{2} + 8 x - 12}{x - 2}$

Divide $x^{3} - 3 x^{2} + 8 x - 12$ by $x - 2$ .

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

$x$

$2$

$x^{3}$

$3 x^{2}$

$8 x$

$12$

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

					$x^{2}$
	$x$	-	$2$		$x^{3}$	-	$3 x^{2}$	+	$8 x$	-	$12$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$3 x^{2}$	+	$8 x$	-	$12$
			+	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$3 x^{2}$	+	$8 x$	-	$12$
			-	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$3 x^{2}$	+	$8 x$	-	$12$
			-	$x^{3}$	+	$2 x^{2}$
					-	$x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$3 x^{2}$	+	$8 x$	-	$12$
			-	$x^{3}$	+	$2 x^{2}$
					-	$x^{2}$	+	$8 x$

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

				$x^{2}$	-	$x$
$x$	-	$2$		$x^{3}$	-	$3 x^{2}$	+	$8 x$	-	$12$
			-	$x^{3}$	+	$2 x^{2}$
					-	$x^{2}$	+	$8 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$x$	-	$2$		$x^{3}$	-	$3 x^{2}$	+	$8 x$	-	$12$
			-	$x^{3}$	+	$2 x^{2}$
					-	$x^{2}$	+	$8 x$
					-	$x^{2}$	+	$2 x$

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

				$x^{2}$	-	$x$
$x$	-	$2$		$x^{3}$	-	$3 x^{2}$	+	$8 x$	-	$12$
			-	$x^{3}$	+	$2 x^{2}$
					-	$x^{2}$	+	$8 x$
					+	$x^{2}$	-	$2 x$

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

				$x^{2}$	-	$x$
$x$	-	$2$		$x^{3}$	-	$3 x^{2}$	+	$8 x$	-	$12$
			-	$x^{3}$	+	$2 x^{2}$
					-	$x^{2}$	+	$8 x$
					+	$x^{2}$	-	$2 x$
							+	$6 x$

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

				$x^{2}$	-	$x$
$x$	-	$2$		$x^{3}$	-	$3 x^{2}$	+	$8 x$	-	$12$
			-	$x^{3}$	+	$2 x^{2}$
					-	$x^{2}$	+	$8 x$
					+	$x^{2}$	-	$2 x$
							+	$6 x$	-	$12$

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

				$x^{2}$	-	$x$	+	$6$
$x$	-	$2$		$x^{3}$	-	$3 x^{2}$	+	$8 x$	-	$12$
			-	$x^{3}$	+	$2 x^{2}$
					-	$x^{2}$	+	$8 x$
					+	$x^{2}$	-	$2 x$
							+	$6 x$	-	$12$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	+	$6$
$x$	-	$2$		$x^{3}$	-	$3 x^{2}$	+	$8 x$	-	$12$
			-	$x^{3}$	+	$2 x^{2}$
					-	$x^{2}$	+	$8 x$
					+	$x^{2}$	-	$2 x$
							+	$6 x$	-	$12$
							+	$6 x$	-	$12$

The expression needs to be subtracted from the dividend, so change all the signs in $6 x - 12$

				$x^{2}$	-	$x$	+	$6$
$x$	-	$2$		$x^{3}$	-	$3 x^{2}$	+	$8 x$	-	$12$
			-	$x^{3}$	+	$2 x^{2}$
					-	$x^{2}$	+	$8 x$
					+	$x^{2}$	-	$2 x$
							+	$6 x$	-	$12$
							-	$6 x$	+	$12$

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

				$x^{2}$	-	$x$	+	$6$
$x$	-	$2$		$x^{3}$	-	$3 x^{2}$	+	$8 x$	-	$12$
			-	$x^{3}$	+	$2 x^{2}$
					-	$x^{2}$	+	$8 x$
					+	$x^{2}$	-	$2 x$
							+	$6 x$	-	$12$
							-	$6 x$	+	$12$
										$0$

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

$x^{2} - x + 6$

Write $x^{3} - 3 x^{2} + 8 x - 12$ as a set of factors.

$(x + 2) (x - 2) ((x - 2) (x^{2} - x + 6))$

Remove unnecessary parentheses.

$(x + 2) (x - 2) (x - 2) (x^{2} - x + 6)$

Step 22

Combine exponents.

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

$(x + 2) ({(x - 2)}^{1} (x - 2)) (x^{2} - x + 6)$

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

$(x + 2) ({(x - 2)}^{1} {(x - 2)}^{1}) (x^{2} - x + 6)$

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

$(x + 2) {(x - 2)}^{1 + 1} (x^{2} - x + 6)$

Add $1$ and $1$ .

$(x + 2) {(x - 2)}^{2} (x^{2} - x + 6)$

Write as a Set of Linear Factors x^5-3x^4+4x^3-32x+48

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