Write as a Set of Linear Factors x^3-111x+110

$x^{3} - 111 x + 110$

Step 1

Factor $x^{3} - 111 x + 110$ 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.

$p = \pm 1, \pm 110, \pm 2, \pm 55, \pm 5, \pm 22, \pm 10, \pm 11 q = \pm 1$

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

$\pm 1, \pm 110, \pm 2, \pm 55, \pm 5, \pm 22, \pm 10, \pm 11$

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.

$1^{3} - 111 \cdot 1 + 110$

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

$1 - 111 \cdot 1 + 110$

Multiply $- 111$ by $1$ .

$1 - 111 + 110$

Subtract $111$ from $1$ .

$- 110 + 110$

Add $- 110$ and $110$ .

$0$

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.

$\frac{x^{3} - 111 x + 110}{x - 1}$

Divide $x^{3} - 111 x + 110$ 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 $0$ .

$x$

$1$

$x^{3}$

$0 x^{2}$

$111 x$

$110$

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$111 x$	+	$110$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$111 x$	+	$110$
			+	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$111 x$	+	$110$
			-	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$111 x$	+	$110$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$111 x$	+	$110$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$111 x$

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

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$111 x$	+	$110$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$111 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$111 x$	+	$110$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$111 x$
					+	$x^{2}$	-	$x$

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

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$111 x$	+	$110$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$111 x$
					-	$x^{2}$	+	$x$

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

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$111 x$	+	$110$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$111 x$
					-	$x^{2}$	+	$x$
							-	$110 x$

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

				$x^{2}$	+	$x$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$111 x$	+	$110$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$111 x$
					-	$x^{2}$	+	$x$
							-	$110 x$	+	$110$

Divide the highest order term in the dividend $- 110 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$x$	-	$110$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$111 x$	+	$110$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$111 x$
					-	$x^{2}$	+	$x$
							-	$110 x$	+	$110$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$	-	$110$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$111 x$	+	$110$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$111 x$
					-	$x^{2}$	+	$x$
							-	$110 x$	+	$110$
							-	$110 x$	+	$110$

The expression needs to be subtracted from the dividend, so change all the signs in $- 110 x + 110$

				$x^{2}$	+	$x$	-	$110$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$111 x$	+	$110$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$111 x$
					-	$x^{2}$	+	$x$
							-	$110 x$	+	$110$
							+	$110 x$	-	$110$

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

				$x^{2}$	+	$x$	-	$110$
$x$	-	$1$		$x^{3}$	+	$0 x^{2}$	-	$111 x$	+	$110$
			-	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	-	$111 x$
					-	$x^{2}$	+	$x$
							-	$110 x$	+	$110$
							+	$110 x$	-	$110$
										$0$

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

$x^{2} + x - 110$

Write $x^{3} - 111 x + 110$ as a set of factors.

$(x - 1) (x^{2} + x - 110)$

Step 2

Factor $x^{2} + x - 110$ 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 $- 110$ and whose sum is $1$ .

$- 10, 11$

Write the factored form using these integers.

$(x - 1) ((x - 10) (x + 11))$

Remove unnecessary parentheses.

$(x - 1) (x - 10) (x + 11)$

Write as a Set of Linear Factors x^3-111x+110

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