Factor -x^5+36x^3-22x^2-147x-90

$- x^{5} + 36 x^{3} - 22 x^{2} - 147 x - 90$

Regroup terms.

$- x^{5} + 36 x^{3} - 22 x^{2} - 147 x - 90$

Factor $- x^{3}$ out of $- x^{5} + 36 x^{3}$ .

Factor $- x^{3}$ out of $- x^{5}$ .

$- x^{3} (x^{2}) + 36 x^{3} - 22 x^{2} - 147 x - 90$

Factor $- x^{3}$ out of $36 x^{3}$ .

$- x^{3} (x^{2}) - x^{3} (- 36) - 22 x^{2} - 147 x - 90$

Factor $- x^{3}$ out of $- x^{3} (x^{2}) - x^{3} (- 36)$ .

$- x^{3} (x^{2} - 36) - 22 x^{2} - 147 x - 90$

Rewrite $36$ as $6^{2}$ .

$- x^{3} (x^{2} - 6^{2}) - 22 x^{2} - 147 x - 90$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

$- x^{3} ((x + 6) (x - 6)) - 22 x^{2} - 147 x - 90$

Remove unnecessary parentheses.

$- x^{3} (x + 6) (x - 6) - 22 x^{2} - 147 x - 90$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 22 \cdot - 90 = 1980$ and whose sum is $b = - 147$ .

Factor $- 147$ out of $- 147 x$ .

$- x^{3} (x + 6) (x - 6) - 22 x^{2} - 147 (x) - 90$

Rewrite $- 147$ as $- 15$ plus $- 132$

$- x^{3} (x + 6) (x - 6) - 22 x^{2} + (- 15 - 132) x - 90$

Apply the distributive property.

$- x^{3} (x + 6) (x - 6) - 22 x^{2} - 15 x - 132 x - 90$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- x^{3} (x + 6) (x - 6) + (- 22 x^{2} - 15 x) - 132 x - 90$

Factor out the greatest common factor (GCF) from each group.

$- x^{3} (x + 6) (x - 6) + x (- 22 x - 15) + 6 (- 22 x - 15)$

Factor the polynomial by factoring out the greatest common factor, $- 22 x - 15$ .

$- x^{3} (x + 6) (x - 6) + (- 22 x - 15) (x + 6)$

Factor $x + 6$ out of $- x^{3} (x + 6) (x - 6) + (- 22 x - 15) (x + 6)$ .

Factor $x + 6$ out of $- x^{3} (x + 6) (x - 6)$ .

$(x + 6) (- x^{3} (x - 6)) + (- 22 x - 15) (x + 6)$

Factor $x + 6$ out of $(- 22 x - 15) (x + 6)$ .

$(x + 6) (- x^{3} (x - 6)) + (x + 6) (- 22 x - 15)$

Factor $x + 6$ out of $(x + 6) (- x^{3} (x - 6)) + (x + 6) (- 22 x - 15)$ .

$(x + 6) (- x^{3} (x - 6) - 22 x - 15)$

Apply the distributive property.

$(x + 6) (- x^{3} x - x^{3} \cdot - 6 - 22 x - 15)$

Multiply $x^{3}$ by $x$ by adding the exponents.

Move $x$ .

$(x + 6) (- (x \cdot x^{3}) - x^{3} \cdot - 6 - 22 x - 15)$

Multiply $x$ by $x^{3}$ .

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

$(x + 6) (- (x^{1} x^{3}) - x^{3} \cdot - 6 - 22 x - 15)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x + 6) (- x^{1 + 3} - x^{3} \cdot - 6 - 22 x - 15)$

Add $1$ and $3$ .

$(x + 6) (- x^{4} - x^{3} \cdot - 6 - 22 x - 15)$

Multiply $- 6$ by $- 1$ .

$(x + 6) (- x^{4} + 6 x^{3} - 22 x - 15)$

Factor.

Rewrite $- x^{4} + 6 x^{3} - 22 x - 15$ in a factored form.

Factor $- x^{4} + 6 x^{3} - 22 x - 15$ 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.

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

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

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

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.

$- {(- 1)}^{4} + 6 {(- 1)}^{3} - 22 \cdot - 1 - 15$

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

$- 1 \cdot 1 + 6 {(- 1)}^{3} - 22 \cdot - 1 - 15$

Multiply $- 1$ by $1$ .

$- 1 + 6 {(- 1)}^{3} - 22 \cdot - 1 - 15$

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

$- 1 + 6 \cdot - 1 - 22 \cdot - 1 - 15$

Multiply $6$ by $- 1$ .

$- 1 - 6 - 22 \cdot - 1 - 15$

Subtract $6$ from $- 1$ .

$- 7 - 22 \cdot - 1 - 15$

Multiply $- 22$ by $- 1$ .

$- 7 + 22 - 15$

Add $- 7$ and $22$ .

$15 - 15$

Subtract $15$ from $15$ .

$0$

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.

$\frac{- x^{4} + 6 x^{3} - 22 x - 15}{x + 1}$

Divide $- x^{4} + 6 x^{3} - 22 x - 15$ 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 $0$ .

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

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

				-	$x^{3}$
	$x$	+	$1$	-	$x^{4}$	+	$6 x^{3}$	+	$0 x^{2}$	-	$22 x$	-	$15$

Multiply the new quotient term by the divisor.

			-	$x^{3}$
$x$	+	$1$	-	$x^{4}$	+	$6 x^{3}$	+	$0 x^{2}$	-	$22 x$	-	$15$
			-	$x^{4}$	-	$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

Divide the highest order term in the dividend $7 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{3} + 7 x^{2}$

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

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

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

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

$x^{3}$

$7 x^{2}$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$22 x$

Divide the highest order term in the dividend $- 7 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$7 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$22 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$7 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$22 x$

$7 x^{2}$

$7 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{2} - 7 x$

$x^{3}$

$7 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$22 x$

$7 x^{2}$

$7 x$

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

$x^{3}$

$7 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$22 x$

$7 x^{2}$

$7 x$

$15 x$

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

$x^{3}$

$7 x^{2}$

$7 x$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$22 x$

$7 x^{2}$

$7 x$

$15 x$

$15$

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

$x^{3}$

$7 x^{2}$

$7 x$

$15$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$22 x$

$7 x^{2}$

$7 x$

$15 x$

$15$

Multiply the new quotient term by the divisor.

$x^{3}$

$7 x^{2}$

$7 x$

$15$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$22 x$

$7 x^{2}$

$7 x$

$15 x$

$15$

$15 x$

$15$

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

$x^{3}$

$7 x^{2}$

$7 x$

$15$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$22 x$

$7 x^{2}$

$7 x$

$15 x$

$15$

$15 x$

$15$

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

$x^{3}$

$7 x^{2}$

$7 x$

$15$

$x$

$1$

$x^{4}$

$6 x^{3}$

$0 x^{2}$

$22 x$

$15$

$x^{4}$

$x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$22 x$

$7 x^{2}$

$7 x$

$15 x$

$15$

$15 x$

$15$

$0$

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

$- x^{3} + 7 x^{2} - 7 x - 15$

Write $- x^{4} + 6 x^{3} - 22 x - 15$ as a set of factors.

$(x + 6) ((x + 1) (- x^{3} + 7 x^{2} - 7 x - 15))$

Factor $- x^{3} + 7 x^{2} - 7 x - 15$ 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.

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

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

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

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.

$- {(- 1)}^{3} + 7 {(- 1)}^{2} - 7 \cdot - 1 - 15$

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

$- - 1 + 7 {(- 1)}^{2} - 7 \cdot - 1 - 15$

Multiply $- 1$ by $- 1$ .

$1 + 7 {(- 1)}^{2} - 7 \cdot - 1 - 15$

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

$1 + 7 \cdot 1 - 7 \cdot - 1 - 15$

Multiply $7$ by $1$ .

$1 + 7 - 7 \cdot - 1 - 15$

Add $1$ and $7$ .

$8 - 7 \cdot - 1 - 15$

Multiply $- 7$ by $- 1$ .

$8 + 7 - 15$

Add $8$ and $7$ .

$15 - 15$

Subtract $15$ from $15$ .

$0$

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.

$\frac{- x^{3} + 7 x^{2} - 7 x - 15}{x + 1}$

Divide $- x^{3} + 7 x^{2} - 7 x - 15$ 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 $0$ .

$x$

$1$

$x^{3}$

$7 x^{2}$

$7 x$

$15$

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

				-	$x^{2}$
	$x$	+	$1$	-	$x^{3}$	+	$7 x^{2}$	-	$7 x$	-	$15$

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	+	$1$	-	$x^{3}$	+	$7 x^{2}$	-	$7 x$	-	$15$
			-	$x^{3}$	-	$x^{2}$

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

			-	$x^{2}$
$x$	+	$1$	-	$x^{3}$	+	$7 x^{2}$	-	$7 x$	-	$15$
			+	$x^{3}$	+	$x^{2}$

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

			-	$x^{2}$
$x$	+	$1$	-	$x^{3}$	+	$7 x^{2}$	-	$7 x$	-	$15$
			+	$x^{3}$	+	$x^{2}$
					+	$8 x^{2}$

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

			-	$x^{2}$
$x$	+	$1$	-	$x^{3}$	+	$7 x^{2}$	-	$7 x$	-	$15$
			+	$x^{3}$	+	$x^{2}$
					+	$8 x^{2}$	-	$7 x$

Divide the highest order term in the dividend $8 x^{2}$ by the highest order term in divisor $x$ .

			-	$x^{2}$	+	$8 x$
$x$	+	$1$	-	$x^{3}$	+	$7 x^{2}$	-	$7 x$	-	$15$
			+	$x^{3}$	+	$x^{2}$
					+	$8 x^{2}$	-	$7 x$

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$8 x$
$x$	+	$1$	-	$x^{3}$	+	$7 x^{2}$	-	$7 x$	-	$15$
			+	$x^{3}$	+	$x^{2}$
					+	$8 x^{2}$	-	$7 x$
					+	$8 x^{2}$	+	$8 x$

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{2} + 8 x$

			-	$x^{2}$	+	$8 x$
$x$	+	$1$	-	$x^{3}$	+	$7 x^{2}$	-	$7 x$	-	$15$
			+	$x^{3}$	+	$x^{2}$
					+	$8 x^{2}$	-	$7 x$
					-	$8 x^{2}$	-	$8 x$

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

			-	$x^{2}$	+	$8 x$
$x$	+	$1$	-	$x^{3}$	+	$7 x^{2}$	-	$7 x$	-	$15$
			+	$x^{3}$	+	$x^{2}$
					+	$8 x^{2}$	-	$7 x$
					-	$8 x^{2}$	-	$8 x$
							-	$15 x$

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

			-	$x^{2}$	+	$8 x$
$x$	+	$1$	-	$x^{3}$	+	$7 x^{2}$	-	$7 x$	-	$15$
			+	$x^{3}$	+	$x^{2}$
					+	$8 x^{2}$	-	$7 x$
					-	$8 x^{2}$	-	$8 x$
							-	$15 x$	-	$15$

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

			-	$x^{2}$	+	$8 x$	-	$15$
$x$	+	$1$	-	$x^{3}$	+	$7 x^{2}$	-	$7 x$	-	$15$
			+	$x^{3}$	+	$x^{2}$
					+	$8 x^{2}$	-	$7 x$
					-	$8 x^{2}$	-	$8 x$
							-	$15 x$	-	$15$

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$8 x$	-	$15$
$x$	+	$1$	-	$x^{3}$	+	$7 x^{2}$	-	$7 x$	-	$15$
			+	$x^{3}$	+	$x^{2}$
					+	$8 x^{2}$	-	$7 x$
					-	$8 x^{2}$	-	$8 x$
							-	$15 x$	-	$15$
							-	$15 x$	-	$15$

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

			-	$x^{2}$	+	$8 x$	-	$15$
$x$	+	$1$	-	$x^{3}$	+	$7 x^{2}$	-	$7 x$	-	$15$
			+	$x^{3}$	+	$x^{2}$
					+	$8 x^{2}$	-	$7 x$
					-	$8 x^{2}$	-	$8 x$
							-	$15 x$	-	$15$
							+	$15 x$	+	$15$

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

			-	$x^{2}$	+	$8 x$	-	$15$
$x$	+	$1$	-	$x^{3}$	+	$7 x^{2}$	-	$7 x$	-	$15$
			+	$x^{3}$	+	$x^{2}$
					+	$8 x^{2}$	-	$7 x$
					-	$8 x^{2}$	-	$8 x$
							-	$15 x$	-	$15$
							+	$15 x$	+	$15$
										$0$

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

$- x^{2} + 8 x - 15$

Write $- x^{3} + 7 x^{2} - 7 x - 15$ as a set of factors.

$(x + 6) ((x + 1) ((x + 1) (- x^{2} + 8 x - 15)))$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 15 = 15$ and whose sum is $b = 8$ .

Factor $8$ out of $8 x$ .

$(x + 6) ((x + 1) ((x + 1) (- x^{2} + 8 (x) - 15)))$

Rewrite $8$ as $3$ plus $5$

$(x + 6) ((x + 1) ((x + 1) (- x^{2} + (3 + 5) x - 15)))$

Apply the distributive property.

$(x + 6) ((x + 1) ((x + 1) (- x^{2} + 3 x + 5 x - 15)))$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x + 6) ((x + 1) ((x + 1) ((- x^{2} + 3 x) + 5 x - 15)))$

Factor out the greatest common factor (GCF) from each group.

$(x + 6) ((x + 1) ((x + 1) (x (- x + 3) - 5 (- x + 3))))$

Factor the polynomial by factoring out the greatest common factor, $- x + 3$ .

$(x + 6) ((x + 1) ((x + 1) ((- x + 3) (x - 5))))$

Remove unnecessary parentheses.

$(x + 6) ((x + 1) ((x + 1) (- x + 3) (x - 5)))$

Remove unnecessary parentheses.

$(x + 6) ((x + 1) (x + 1) (- x + 3) (x - 5))$

Combine like factors.

Raise $x + 1$ to the power of $1$ .

$(x + 6) ({(x + 1)}^{1} (x + 1) (- x + 3) (x - 5))$

Raise $x + 1$ to the power of $1$ .

$(x + 6) ({(x + 1)}^{1} {(x + 1)}^{1} (- x + 3) (x - 5))$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x + 6) ({(x + 1)}^{1 + 1} (- x + 3) (x - 5))$

Add $1$ and $1$ .

$(x + 6) ({(x + 1)}^{2} (- x + 3) (x - 5))$

Remove unnecessary parentheses.

$(x + 6) {(x + 1)}^{2} (- x + 3) (x - 5)$

Factor.

Factor $- 1$ out of $- x + 3$ .

Factor $- 1$ out of $- x$ .

$(x + 6) {(x + 1)}^{2} (- (x) + 3) (x - 5)$

Rewrite $3$ as $- 1 (- 3)$ .

$(x + 6) {(x + 1)}^{2} (- (x) - 1 (- 3)) (x - 5)$

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

$(x + 6) {(x + 1)}^{2} (- (x - 3)) (x - 5)$

Remove unnecessary parentheses.

$(x + 6) {(x + 1)}^{2} \cdot - 1 (x - 3) (x - 5)$

Factor.

Factor out negative.

$- (((x + 6) {(x + 1)}^{2} (x - 3)) (x - 5))$

Remove unnecessary parentheses.

$- (x + 6) {(x + 1)}^{2} (x - 3) (x - 5)$

Factor -x^5+36x^3-22x^2-147x-90

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