Factor x^3+9x^2+23x+15

$x^{3} + 9 x^{2} + 23 x + 15$

Step 1

Factor $x^{3} + 9 x^{2} + 23 x + 15$ 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.

$p = \pm 1, \pm 15, \pm 3, \pm 5 q = \pm 1$

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

$\pm 1, \pm 15, \pm 3, \pm 5$

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.

${(- 1)}^{3} + 9 {(- 1)}^{2} + 23 \cdot - 1 + 15$

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

$- 1 + 9 {(- 1)}^{2} + 23 \cdot - 1 + 15$

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

$- 1 + 9 \cdot 1 + 23 \cdot - 1 + 15$

Multiply $9$ by $1$ .

$- 1 + 9 + 23 \cdot - 1 + 15$

Add $- 1$ and $9$ .

$8 + 23 \cdot - 1 + 15$

Multiply $23$ by $- 1$ .

$8 - 23 + 15$

Subtract $23$ from $8$ .

$- 15 + 15$

Add $- 15$ and $15$ .

$0$

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.

$\frac{x^{3} + 9 x^{2} + 23 x + 15}{x + 1}$

Divide $x^{3} + 9 x^{2} + 23 x + 15$ 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 $0$ .

$x$

$1$

$x^{3}$

$9 x^{2}$

$23 x$

$15$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$23 x$	+	$15$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$23 x$	+	$15$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$23 x$	+	$15$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$23 x$	+	$15$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$23 x$	+	$15$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$23 x$

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

				$x^{2}$	+	$8 x$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$23 x$	+	$15$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$23 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$8 x$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$23 x$	+	$15$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$23 x$
					+	$8 x^{2}$	+	$8 x$

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

				$x^{2}$	+	$8 x$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$23 x$	+	$15$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$23 x$
					-	$8 x^{2}$	-	$8 x$

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

				$x^{2}$	+	$8 x$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$23 x$	+	$15$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$23 x$
					-	$8 x^{2}$	-	$8 x$
							+	$15 x$

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

				$x^{2}$	+	$8 x$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$23 x$	+	$15$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$23 x$
					-	$8 x^{2}$	-	$8 x$
							+	$15 x$	+	$15$

Divide the highest order term in the dividend $15 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$8 x$	+	$15$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$23 x$	+	$15$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$23 x$
					-	$8 x^{2}$	-	$8 x$
							+	$15 x$	+	$15$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$8 x$	+	$15$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$23 x$	+	$15$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$23 x$
					-	$8 x^{2}$	-	$8 x$
							+	$15 x$	+	$15$
							+	$15 x$	+	$15$

The expression needs to be subtracted from the dividend, so change all the signs in $15 x + 15$

				$x^{2}$	+	$8 x$	+	$15$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$23 x$	+	$15$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$23 x$
					-	$8 x^{2}$	-	$8 x$
							+	$15 x$	+	$15$
							-	$15 x$	-	$15$

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

				$x^{2}$	+	$8 x$	+	$15$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$23 x$	+	$15$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$23 x$
					-	$8 x^{2}$	-	$8 x$
							+	$15 x$	+	$15$
							-	$15 x$	-	$15$
										$0$

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

$x^{2} + 8 x + 15$

Write $x^{3} + 9 x^{2} + 23 x + 15$ as a set of factors.

$(x + 1) (x^{2} + 8 x + 15)$

Step 2

Factor $x^{2} + 8 x + 15$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $15$ and whose sum is $8$ .

$3, 5$

Write the factored form using these integers.

$(x + 1) ((x + 3) (x + 5))$

Remove unnecessary parentheses.

$(x + 1) (x + 3) (x + 5)$

Factor x^3+9x^2+23x+15

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