Write as a Set of Linear Factors x^3-7x^2+8x+16

$x^{3} - 7 x^{2} + 8 x + 16$

Step 1

Factor $x^{3} - 7 x^{2} + 8 x + 16$ 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.

$p = \pm 1, \pm 16, \pm 2, \pm 8, \pm 4 q = \pm 1$

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

$\pm 1, \pm 16, \pm 2, \pm 8, \pm 4$

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} - 7 {(- 1)}^{2} + 8 \cdot - 1 + 16$

Raise $- 1$ to the power of $3$ .

$- 1 - 7 {(- 1)}^{2} + 8 \cdot - 1 + 16$

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

$- 1 - 7 \cdot 1 + 8 \cdot - 1 + 16$

Multiply $- 7$ by $1$ .

$- 1 - 7 + 8 \cdot - 1 + 16$

Subtract $7$ from $- 1$ .

$- 8 + 8 \cdot - 1 + 16$

Multiply $8$ by $- 1$ .

$- 8 - 8 + 16$

Subtract $8$ from $- 8$ .

$- 16 + 16$

Add $- 16$ and $16$ .

$0$

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

$\frac{x^{3} - 7 x^{2} + 8 x + 16}{x + 1}$

Divide $x^{3} - 7 x^{2} + 8 x + 16$ by $x + 1$ .

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

$x$

$1$

$x^{3}$

$7 x^{2}$

$8 x$

$16$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$8 x$	+	$16$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$8 x$	+	$16$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$8 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$8 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$8 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$8 x$

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

				$x^{2}$	-	$8 x$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$8 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$8 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$8 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$8 x$
					-	$8 x^{2}$	-	$8 x$

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

				$x^{2}$	-	$8 x$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$8 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$8 x$
					+	$8 x^{2}$	+	$8 x$

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

				$x^{2}$	-	$8 x$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$8 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$8 x$
					+	$8 x^{2}$	+	$8 x$
							+	$16 x$

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

				$x^{2}$	-	$8 x$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$8 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$8 x$
					+	$8 x^{2}$	+	$8 x$
							+	$16 x$	+	$16$

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

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$8 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$8 x$
					+	$8 x^{2}$	+	$8 x$
							+	$16 x$	+	$16$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$8 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$8 x$
					+	$8 x^{2}$	+	$8 x$
							+	$16 x$	+	$16$
							+	$16 x$	+	$16$

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

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$8 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$8 x$
					+	$8 x^{2}$	+	$8 x$
							+	$16 x$	+	$16$
							-	$16 x$	-	$16$

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

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$8 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$8 x$
					+	$8 x^{2}$	+	$8 x$
							+	$16 x$	+	$16$
							-	$16 x$	-	$16$
										$0$

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

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

Write $x^{3} - 7 x^{2} + 8 x + 16$ as a set of factors.

$(x + 1) (x^{2} - 8 x + 16)$

Step 2

Factor using the perfect square rule.

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

$(x + 1) (x^{2} - 8 x + 4^{2})$

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

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

Rewrite the polynomial.

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

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

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

Write as a Set of Linear Factors x^3-7x^2+8x+16

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