Write as a Set of Linear Factors x^4-10x^3-70x^2-110x-51

$x^{4} - 10 x^{3} - 70 x^{2} - 110 x - 51$

Factor $x^{4} - 10 x^{3} - 70 x^{2} - 110 x - 51$ 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.

$p = \pm 1, \pm 51, \pm 3, \pm 17 q = \pm 1$

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

$\pm 1, \pm 51, \pm 3, \pm 17$

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.

${(- 1)}^{4} - 10 {(- 1)}^{3} - 70 {(- 1)}^{2} - 110 \cdot - 1 - 51$

Raise $- 1$ to the power of $4$ .

$1 - 10 {(- 1)}^{3} - 70 {(- 1)}^{2} - 110 \cdot - 1 - 51$

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

$1 - 10 \cdot - 1 - 70 {(- 1)}^{2} - 110 \cdot - 1 - 51$

Multiply $- 10$ by $- 1$ .

$1 + 10 - 70 {(- 1)}^{2} - 110 \cdot - 1 - 51$

Add $1$ and $10$ .

$11 - 70 {(- 1)}^{2} - 110 \cdot - 1 - 51$

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

$11 - 70 \cdot 1 - 110 \cdot - 1 - 51$

Multiply $- 70$ by $1$ .

$11 - 70 - 110 \cdot - 1 - 51$

Subtract $70$ from $11$ .

$- 59 - 110 \cdot - 1 - 51$

Multiply $- 110$ by $- 1$ .

$- 59 + 110 - 51$

Add $- 59$ and $110$ .

$51 - 51$

Subtract $51$ from $51$ .

$0$

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.

$\frac{x^{4} - 10 x^{3} - 70 x^{2} - 110 x - 51}{x + 1}$

Divide $x^{4} - 10 x^{3} - 70 x^{2} - 110 x - 51$ 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 $0$ .

$x$

$1$

$x^{4}$

$10 x^{3}$

$70 x^{2}$

$110 x$

$51$

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

					$x^{3}$
	$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$10 x^{3}$

$70 x^{2}$

$110 x$

$51$

$x^{4}$

$x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + x^{3}$

				$x^{3}$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$

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

				$x^{3}$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$
					-	$11 x^{3}$

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

				$x^{3}$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$
					-	$11 x^{3}$	-	$70 x^{2}$

Divide the highest order term in the dividend $- 11 x^{3}$ by the highest order term in divisor $x$ .

				$x^{3}$	-	$11 x^{2}$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$
					-	$11 x^{3}$	-	$70 x^{2}$

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$11 x^{2}$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$
					-	$11 x^{3}$	-	$70 x^{2}$
					-	$11 x^{3}$	-	$11 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 11 x^{3} - 11 x^{2}$

				$x^{3}$	-	$11 x^{2}$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$
					-	$11 x^{3}$	-	$70 x^{2}$
					+	$11 x^{3}$	+	$11 x^{2}$

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

				$x^{3}$	-	$11 x^{2}$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$
					-	$11 x^{3}$	-	$70 x^{2}$
					+	$11 x^{3}$	+	$11 x^{2}$
							-	$59 x^{2}$

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

				$x^{3}$	-	$11 x^{2}$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$
					-	$11 x^{3}$	-	$70 x^{2}$
					+	$11 x^{3}$	+	$11 x^{2}$
							-	$59 x^{2}$	-	$110 x$

Divide the highest order term in the dividend $- 59 x^{2}$ by the highest order term in divisor $x$ .

				$x^{3}$	-	$11 x^{2}$	-	$59 x$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$
					-	$11 x^{3}$	-	$70 x^{2}$
					+	$11 x^{3}$	+	$11 x^{2}$
							-	$59 x^{2}$	-	$110 x$

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$11 x^{2}$	-	$59 x$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$
					-	$11 x^{3}$	-	$70 x^{2}$
					+	$11 x^{3}$	+	$11 x^{2}$
							-	$59 x^{2}$	-	$110 x$
							-	$59 x^{2}$	-	$59 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 59 x^{2} - 59 x$

$x^{3}$

$11 x^{2}$

$59 x$

$x$

$1$

$x^{4}$

$10 x^{3}$

$70 x^{2}$

$110 x$

$51$

$x^{4}$

$x^{3}$

$11 x^{3}$

$70 x^{2}$

$11 x^{3}$

$11 x^{2}$

$59 x^{2}$

$110 x$

$59 x^{2}$

$59 x$

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

$x^{3}$

$11 x^{2}$

$59 x$

$x$

$1$

$x^{4}$

$10 x^{3}$

$70 x^{2}$

$110 x$

$51$

$x^{4}$

$x^{3}$

$11 x^{3}$

$70 x^{2}$

$11 x^{3}$

$11 x^{2}$

$59 x^{2}$

$110 x$

$59 x^{2}$

$59 x$

$51 x$

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

				$x^{3}$	-	$11 x^{2}$	-	$59 x$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$
					-	$11 x^{3}$	-	$70 x^{2}$
					+	$11 x^{3}$	+	$11 x^{2}$
							-	$59 x^{2}$	-	$110 x$
							+	$59 x^{2}$	+	$59 x$
									-	$51 x$	-	$51$

Divide the highest order term in the dividend $- 51 x$ by the highest order term in divisor $x$ .

				$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$
					-	$11 x^{3}$	-	$70 x^{2}$
					+	$11 x^{3}$	+	$11 x^{2}$
							-	$59 x^{2}$	-	$110 x$
							+	$59 x^{2}$	+	$59 x$
									-	$51 x$	-	$51$

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$
					-	$11 x^{3}$	-	$70 x^{2}$
					+	$11 x^{3}$	+	$11 x^{2}$
							-	$59 x^{2}$	-	$110 x$
							+	$59 x^{2}$	+	$59 x$
									-	$51 x$	-	$51$
									-	$51 x$	-	$51$

The expression needs to be subtracted from the dividend, so change all the signs in $- 51 x - 51$

				$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$
					-	$11 x^{3}$	-	$70 x^{2}$
					+	$11 x^{3}$	+	$11 x^{2}$
							-	$59 x^{2}$	-	$110 x$
							+	$59 x^{2}$	+	$59 x$
									-	$51 x$	-	$51$
									+	$51 x$	+	$51$

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

				$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
$x$	+	$1$		$x^{4}$	-	$10 x^{3}$	-	$70 x^{2}$	-	$110 x$	-	$51$
			-	$x^{4}$	-	$x^{3}$
					-	$11 x^{3}$	-	$70 x^{2}$
					+	$11 x^{3}$	+	$11 x^{2}$
							-	$59 x^{2}$	-	$110 x$
							+	$59 x^{2}$	+	$59 x$
									-	$51 x$	-	$51$
									+	$51 x$	+	$51$
												$0$

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

$x^{3} - 11 x^{2} - 59 x - 51$

Write $x^{4} - 10 x^{3} - 70 x^{2} - 110 x - 51$ as a set of factors.

$(x + 1) (x^{3} - 11 x^{2} - 59 x - 51)$

Factor $x^{3} - 11 x^{2} - 59 x - 51$ 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.

$p = \pm 1, \pm 51, \pm 3, \pm 17 q = \pm 1$

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

$\pm 1, \pm 51, \pm 3, \pm 17$

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.

${(- 3)}^{3} - 11 {(- 3)}^{2} - 59 \cdot - 3 - 51$

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

$- 27 - 11 {(- 3)}^{2} - 59 \cdot - 3 - 51$

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

$- 27 - 11 \cdot 9 - 59 \cdot - 3 - 51$

Multiply $- 11$ by $9$ .

$- 27 - 99 - 59 \cdot - 3 - 51$

Subtract $99$ from $- 27$ .

$- 126 - 59 \cdot - 3 - 51$

Multiply $- 59$ by $- 3$ .

$- 126 + 177 - 51$

Add $- 126$ and $177$ .

$51 - 51$

Subtract $51$ from $51$ .

$0$

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.

$\frac{x^{3} - 11 x^{2} - 59 x - 51}{x + 3}$

Divide $x^{3} - 11 x^{2} - 59 x - 51$ 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 $0$ .

$x$

$3$

$x^{3}$

$11 x^{2}$

$59 x$

$51$

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

					$x^{2}$
	$x$	+	$3$		$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
			+	$x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
			-	$x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
			-	$x^{3}$	-	$3 x^{2}$
					-	$14 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
			-	$x^{3}$	-	$3 x^{2}$
					-	$14 x^{2}$	-	$59 x$

Divide the highest order term in the dividend $- 14 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$14 x$
$x$	+	$3$		$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
			-	$x^{3}$	-	$3 x^{2}$
					-	$14 x^{2}$	-	$59 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$14 x$
$x$	+	$3$		$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
			-	$x^{3}$	-	$3 x^{2}$
					-	$14 x^{2}$	-	$59 x$
					-	$14 x^{2}$	-	$42 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 14 x^{2} - 42 x$

				$x^{2}$	-	$14 x$
$x$	+	$3$		$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
			-	$x^{3}$	-	$3 x^{2}$
					-	$14 x^{2}$	-	$59 x$
					+	$14 x^{2}$	+	$42 x$

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

				$x^{2}$	-	$14 x$
$x$	+	$3$		$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
			-	$x^{3}$	-	$3 x^{2}$
					-	$14 x^{2}$	-	$59 x$
					+	$14 x^{2}$	+	$42 x$
							-	$17 x$

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

				$x^{2}$	-	$14 x$
$x$	+	$3$		$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
			-	$x^{3}$	-	$3 x^{2}$
					-	$14 x^{2}$	-	$59 x$
					+	$14 x^{2}$	+	$42 x$
							-	$17 x$	-	$51$

Divide the highest order term in the dividend $- 17 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$14 x$	-	$17$
$x$	+	$3$		$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
			-	$x^{3}$	-	$3 x^{2}$
					-	$14 x^{2}$	-	$59 x$
					+	$14 x^{2}$	+	$42 x$
							-	$17 x$	-	$51$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$14 x$	-	$17$
$x$	+	$3$		$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
			-	$x^{3}$	-	$3 x^{2}$
					-	$14 x^{2}$	-	$59 x$
					+	$14 x^{2}$	+	$42 x$
							-	$17 x$	-	$51$
							-	$17 x$	-	$51$

The expression needs to be subtracted from the dividend, so change all the signs in $- 17 x - 51$

				$x^{2}$	-	$14 x$	-	$17$
$x$	+	$3$		$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
			-	$x^{3}$	-	$3 x^{2}$
					-	$14 x^{2}$	-	$59 x$
					+	$14 x^{2}$	+	$42 x$
							-	$17 x$	-	$51$
							+	$17 x$	+	$51$

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

				$x^{2}$	-	$14 x$	-	$17$
$x$	+	$3$		$x^{3}$	-	$11 x^{2}$	-	$59 x$	-	$51$
			-	$x^{3}$	-	$3 x^{2}$
					-	$14 x^{2}$	-	$59 x$
					+	$14 x^{2}$	+	$42 x$
							-	$17 x$	-	$51$
							+	$17 x$	+	$51$
										$0$

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

$x^{2} - 14 x - 17$

Write $x^{3} - 11 x^{2} - 59 x - 51$ as a set of factors.

$(x + 1) ((x + 3) (x^{2} - 14 x - 17))$

Remove unnecessary parentheses.

$(x + 1) (x + 3) (x^{2} - 14 x - 17)$

Write as a Set of Linear Factors x^4-10x^3-70x^2-110x-51

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