Factor x^3-9x^2+11x+21

$x^{3} - 9 x^{2} + 11 x + 21$

Step 1

Factor $x^{3} - 9 x^{2} + 11 x + 21$ 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.

$p = \pm 1, \pm 21, \pm 3, \pm 7 q = \pm 1$

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

$\pm 1, \pm 21, \pm 3, \pm 7$

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.

${(- 1)}^{3} - 9 {(- 1)}^{2} + 11 \cdot - 1 + 21$

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

$- 1 - 9 {(- 1)}^{2} + 11 \cdot - 1 + 21$

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

$- 1 - 9 \cdot 1 + 11 \cdot - 1 + 21$

Multiply $- 9$ by $1$ .

$- 1 - 9 + 11 \cdot - 1 + 21$

Subtract $9$ from $- 1$ .

$- 10 + 11 \cdot - 1 + 21$

Multiply $11$ by $- 1$ .

$- 10 - 11 + 21$

Subtract $11$ from $- 10$ .

$- 21 + 21$

Add $- 21$ and $21$ .

$0$

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.

$\frac{x^{3} - 9 x^{2} + 11 x + 21}{x + 1}$

Divide $x^{3} - 9 x^{2} + 11 x + 21$ 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 $0$ .

$x$

$1$

$x^{3}$

$9 x^{2}$

$11 x$

$21$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$9 x^{2}$	+	$11 x$	+	$21$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$9 x^{2}$	+	$11 x$	+	$21$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$9 x^{2}$	+	$11 x$	+	$21$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$9 x^{2}$	+	$11 x$	+	$21$
			-	$x^{3}$	-	$x^{2}$
					-	$10 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$9 x^{2}$	+	$11 x$	+	$21$
			-	$x^{3}$	-	$x^{2}$
					-	$10 x^{2}$	+	$11 x$

Divide the highest order term in the dividend $- 10 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$10 x$
$x$	+	$1$		$x^{3}$	-	$9 x^{2}$	+	$11 x$	+	$21$
			-	$x^{3}$	-	$x^{2}$
					-	$10 x^{2}$	+	$11 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$
$x$	+	$1$		$x^{3}$	-	$9 x^{2}$	+	$11 x$	+	$21$
			-	$x^{3}$	-	$x^{2}$
					-	$10 x^{2}$	+	$11 x$
					-	$10 x^{2}$	-	$10 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} - 10 x$

				$x^{2}$	-	$10 x$
$x$	+	$1$		$x^{3}$	-	$9 x^{2}$	+	$11 x$	+	$21$
			-	$x^{3}$	-	$x^{2}$
					-	$10 x^{2}$	+	$11 x$
					+	$10 x^{2}$	+	$10 x$

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

				$x^{2}$	-	$10 x$
$x$	+	$1$		$x^{3}$	-	$9 x^{2}$	+	$11 x$	+	$21$
			-	$x^{3}$	-	$x^{2}$
					-	$10 x^{2}$	+	$11 x$
					+	$10 x^{2}$	+	$10 x$
							+	$21 x$

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

				$x^{2}$	-	$10 x$
$x$	+	$1$		$x^{3}$	-	$9 x^{2}$	+	$11 x$	+	$21$
			-	$x^{3}$	-	$x^{2}$
					-	$10 x^{2}$	+	$11 x$
					+	$10 x^{2}$	+	$10 x$
							+	$21 x$	+	$21$

Divide the highest order term in the dividend $21 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$10 x$	+	$21$
$x$	+	$1$		$x^{3}$	-	$9 x^{2}$	+	$11 x$	+	$21$
			-	$x^{3}$	-	$x^{2}$
					-	$10 x^{2}$	+	$11 x$
					+	$10 x^{2}$	+	$10 x$
							+	$21 x$	+	$21$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$	+	$21$
$x$	+	$1$		$x^{3}$	-	$9 x^{2}$	+	$11 x$	+	$21$
			-	$x^{3}$	-	$x^{2}$
					-	$10 x^{2}$	+	$11 x$
					+	$10 x^{2}$	+	$10 x$
							+	$21 x$	+	$21$
							+	$21 x$	+	$21$

The expression needs to be subtracted from the dividend, so change all the signs in $21 x + 21$

				$x^{2}$	-	$10 x$	+	$21$
$x$	+	$1$		$x^{3}$	-	$9 x^{2}$	+	$11 x$	+	$21$
			-	$x^{3}$	-	$x^{2}$
					-	$10 x^{2}$	+	$11 x$
					+	$10 x^{2}$	+	$10 x$
							+	$21 x$	+	$21$
							-	$21 x$	-	$21$

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

				$x^{2}$	-	$10 x$	+	$21$
$x$	+	$1$		$x^{3}$	-	$9 x^{2}$	+	$11 x$	+	$21$
			-	$x^{3}$	-	$x^{2}$
					-	$10 x^{2}$	+	$11 x$
					+	$10 x^{2}$	+	$10 x$
							+	$21 x$	+	$21$
							-	$21 x$	-	$21$
										$0$

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

$x^{2} - 10 x + 21$

Write $x^{3} - 9 x^{2} + 11 x + 21$ as a set of factors.

$(x + 1) (x^{2} - 10 x + 21)$

Step 2

Factor $x^{2} - 10 x + 21$ 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 $21$ and whose sum is $- 10$ .

$- 7, - 3$

Write the factored form using these integers.

$(x + 1) ((x - 7) (x - 3))$

Remove unnecessary parentheses.

$(x + 1) (x - 7) (x - 3)$

Factor x^3-9x^2+11x+21

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