Factor x^3+2x^2-13x+10

$x^{3} + 2 x^{2} - 13 x + 10$

Step 1

Factor $x^{3} + 2 x^{2} - 13 x + 10$ 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.

$p = \pm 1, \pm 10, \pm 2, \pm 5 q = \pm 1$

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

$\pm 1, \pm 10, \pm 2, \pm 5$

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.

$1^{3} + 2 \cdot 1^{2} - 13 \cdot 1 + 10$

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

$1 + 2 \cdot 1^{2} - 13 \cdot 1 + 10$

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

$1 + 2 \cdot 1 - 13 \cdot 1 + 10$

Multiply $2$ by $1$ .

$1 + 2 - 13 \cdot 1 + 10$

Add $1$ and $2$ .

$3 - 13 \cdot 1 + 10$

Multiply $- 13$ by $1$ .

$3 - 13 + 10$

Subtract $13$ from $3$ .

$- 10 + 10$

Add $- 10$ and $10$ .

$0$

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.

$\frac{x^{3} + 2 x^{2} - 13 x + 10}{x - 1}$

Divide $x^{3} + 2 x^{2} - 13 x + 10$ 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 $0$ .

$x$

$1$

$x^{3}$

$2 x^{2}$

$13 x$

$10$

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	+	$2 x^{2}$	-	$13 x$	+	$10$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$2 x^{2}$	-	$13 x$	+	$10$
			+	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$2 x^{2}$	-	$13 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$2 x^{2}$	-	$13 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					+	$3 x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$2 x^{2}$	-	$13 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	-	$13 x$

Divide the highest order term in the dividend $3 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$3 x$
$x$	-	$1$		$x^{3}$	+	$2 x^{2}$	-	$13 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	-	$13 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$
$x$	-	$1$		$x^{3}$	+	$2 x^{2}$	-	$13 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	-	$13 x$
					+	$3 x^{2}$	-	$3 x$

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

				$x^{2}$	+	$3 x$
$x$	-	$1$		$x^{3}$	+	$2 x^{2}$	-	$13 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	-	$13 x$
					-	$3 x^{2}$	+	$3 x$

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

				$x^{2}$	+	$3 x$
$x$	-	$1$		$x^{3}$	+	$2 x^{2}$	-	$13 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	-	$13 x$
					-	$3 x^{2}$	+	$3 x$
							-	$10 x$

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

				$x^{2}$	+	$3 x$
$x$	-	$1$		$x^{3}$	+	$2 x^{2}$	-	$13 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	-	$13 x$
					-	$3 x^{2}$	+	$3 x$
							-	$10 x$	+	$10$

Divide the highest order term in the dividend $- 10 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$3 x$	-	$10$
$x$	-	$1$		$x^{3}$	+	$2 x^{2}$	-	$13 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	-	$13 x$
					-	$3 x^{2}$	+	$3 x$
							-	$10 x$	+	$10$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$	-	$10$
$x$	-	$1$		$x^{3}$	+	$2 x^{2}$	-	$13 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	-	$13 x$
					-	$3 x^{2}$	+	$3 x$
							-	$10 x$	+	$10$
							-	$10 x$	+	$10$

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x + 10$

				$x^{2}$	+	$3 x$	-	$10$
$x$	-	$1$		$x^{3}$	+	$2 x^{2}$	-	$13 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	-	$13 x$
					-	$3 x^{2}$	+	$3 x$
							-	$10 x$	+	$10$
							+	$10 x$	-	$10$

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

				$x^{2}$	+	$3 x$	-	$10$
$x$	-	$1$		$x^{3}$	+	$2 x^{2}$	-	$13 x$	+	$10$
			-	$x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	-	$13 x$
					-	$3 x^{2}$	+	$3 x$
							-	$10 x$	+	$10$
							+	$10 x$	-	$10$
										$0$

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

$x^{2} + 3 x - 10$

Write $x^{3} + 2 x^{2} - 13 x + 10$ as a set of factors.

$(x - 1) (x^{2} + 3 x - 10)$

Step 2

Factor $x^{2} + 3 x - 10$ 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 $- 10$ and whose sum is $3$ .

$- 2, 5$

Write the factored form using these integers.

$(x - 1) ((x - 2) (x + 5))$

Remove unnecessary parentheses.

$(x - 1) (x - 2) (x + 5)$

Factor x^3+2x^2-13x+10

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