Write as a Set of Linear Factors x^3+x^2-21x-45

$x^{3} + x^{2} - 21 x - 45$

Factor $x^{3} + x^{2} - 21 x - 45$ 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.

$p = \pm 1, \pm 45, \pm 3, \pm 15, \pm 5, \pm 9 q = \pm 1$

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

$\pm 1, \pm 45, \pm 3, \pm 15, \pm 5, \pm 9$

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

Substitute $- 3$ into the polynomial.

${(- 3)}^{3} + {(- 3)}^{2} - 21 \cdot - 3 - 45$

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

$- 27 + {(- 3)}^{2} - 21 \cdot - 3 - 45$

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

$- 27 + 9 - 21 \cdot - 3 - 45$

Add $- 27$ and $9$ .

$- 18 - 21 \cdot - 3 - 45$

Multiply $- 21$ by $- 3$ .

$- 18 + 63 - 45$

Add $- 18$ and $63$ .

$45 - 45$

Subtract $45$ from $45$ .

$0$

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.

$\frac{x^{3} + x^{2} - 21 x - 45}{x + 3}$

Divide $x^{3} + x^{2} - 21 x - 45$ by $x + 3$ .

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

$x$

$3$

$x^{3}$

$x^{2}$

$21 x$

$45$

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

					$x^{2}$
	$x$	+	$3$		$x^{3}$	+	$x^{2}$	-	$21 x$	-	$45$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$x^{2}$	-	$21 x$	-	$45$
			+	$x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$x^{2}$	-	$21 x$	-	$45$
			-	$x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$x^{2}$	-	$21 x$	-	$45$
			-	$x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$x^{2}$	-	$21 x$	-	$45$
			-	$x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$21 x$

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

				$x^{2}$	-	$2 x$
$x$	+	$3$		$x^{3}$	+	$x^{2}$	-	$21 x$	-	$45$
			-	$x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$21 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$
$x$	+	$3$		$x^{3}$	+	$x^{2}$	-	$21 x$	-	$45$
			-	$x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$21 x$
					-	$2 x^{2}$	-	$6 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} - 6 x$

				$x^{2}$	-	$2 x$
$x$	+	$3$		$x^{3}$	+	$x^{2}$	-	$21 x$	-	$45$
			-	$x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$21 x$
					+	$2 x^{2}$	+	$6 x$

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

				$x^{2}$	-	$2 x$
$x$	+	$3$		$x^{3}$	+	$x^{2}$	-	$21 x$	-	$45$
			-	$x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$21 x$
					+	$2 x^{2}$	+	$6 x$
							-	$15 x$

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

				$x^{2}$	-	$2 x$
$x$	+	$3$		$x^{3}$	+	$x^{2}$	-	$21 x$	-	$45$
			-	$x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$21 x$
					+	$2 x^{2}$	+	$6 x$
							-	$15 x$	-	$45$

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

				$x^{2}$	-	$2 x$	-	$15$
$x$	+	$3$		$x^{3}$	+	$x^{2}$	-	$21 x$	-	$45$
			-	$x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$21 x$
					+	$2 x^{2}$	+	$6 x$
							-	$15 x$	-	$45$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$	-	$15$
$x$	+	$3$		$x^{3}$	+	$x^{2}$	-	$21 x$	-	$45$
			-	$x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$21 x$
					+	$2 x^{2}$	+	$6 x$
							-	$15 x$	-	$45$
							-	$15 x$	-	$45$

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x - 45$

				$x^{2}$	-	$2 x$	-	$15$
$x$	+	$3$		$x^{3}$	+	$x^{2}$	-	$21 x$	-	$45$
			-	$x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$21 x$
					+	$2 x^{2}$	+	$6 x$
							-	$15 x$	-	$45$
							+	$15 x$	+	$45$

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

				$x^{2}$	-	$2 x$	-	$15$
$x$	+	$3$		$x^{3}$	+	$x^{2}$	-	$21 x$	-	$45$
			-	$x^{3}$	-	$3 x^{2}$
					-	$2 x^{2}$	-	$21 x$
					+	$2 x^{2}$	+	$6 x$
							-	$15 x$	-	$45$
							+	$15 x$	+	$45$
										$0$

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

$x^{2} - 2 x - 15$

Write $x^{3} + x^{2} - 21 x - 45$ as a set of factors.

$(x + 3) (x^{2} - 2 x - 15)$

Factor $x^{2} - 2 x - 15$ 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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Write the factored form using these integers.

$(x + 3) ((x - 5) (x + 3))$

Remove unnecessary parentheses.

$(x + 3) (x - 5) (x + 3)$

Write as a Set of Linear Factors x^3+x^2-21x-45

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