Find the Roots (Zeros) x^5-3x^4-8x^3-8x^2-9x-5=0

$x^{5} - 3 x^{4} - 8 x^{3} - 8 x^{2} - 9 x - 5 = 0$

Factor the left side of the equation.

Regroup terms.

$x^{5} - 8 x^{2} - 9 x - 3 x^{4} - 8 x^{3} - 5 = 0$

Factor $x$ out of $x^{5} - 8 x^{2} - 9 x$ .

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

$x \cdot x^{4} - 8 x^{2} - 9 x - 3 x^{4} - 8 x^{3} - 5 = 0$

Factor $x$ out of $- 8 x^{2}$ .

$x \cdot x^{4} + x (- 8 x) - 9 x - 3 x^{4} - 8 x^{3} - 5 = 0$

Factor $x$ out of $- 9 x$ .

$x \cdot x^{4} + x (- 8 x) + x \cdot - 9 - 3 x^{4} - 8 x^{3} - 5 = 0$

Factor $x$ out of $x \cdot x^{4} + x (- 8 x)$ .

$x (x^{4} - 8 x) + x \cdot - 9 - 3 x^{4} - 8 x^{3} - 5 = 0$

Factor $x$ out of $x (x^{4} - 8 x) + x \cdot - 9$ .

$x (x^{4} - 8 x - 9) - 3 x^{4} - 8 x^{3} - 5 = 0$

Factor.

Factor $x^{4} - 8 x - 9$ 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.

$p = \pm 1, \pm 9, \pm 3 q = \pm 1$

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

$\pm 1, \pm 9, \pm 3$

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.

${(- 1)}^{4} - 8 \cdot - 1 - 9$

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

$1 - 8 \cdot - 1 - 9$

Multiply $- 8$ by $- 1$ .

$1 + 8 - 9$

Add $1$ and $8$ .

$9 - 9$

Subtract $9$ from $9$ .

$0$

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.

$\frac{x^{4} - 8 x - 9}{x + 1}$

Divide $x^{4} - 8 x - 9$ 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 $0$ .

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

$0 x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$8 x$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$8 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$8 x$

$x^{2}$

$x$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$8 x$

$x^{2}$

$x$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$8 x$

$x^{2}$

$x$

$9 x$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$8 x$

$x^{2}$

$x$

$9 x$

$9$

Divide the highest order term in the dividend $- 9 x$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2}$

$x$

$9$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$8 x$

$x^{2}$

$x$

$9 x$

$9$

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$x^{2}$	+	$x$	-	$9$
$x$	+	$1$		$x^{4}$	+	$0 x^{3}$	+	$0 x^{2}$	-	$8 x$	-	$9$
			-	$x^{4}$	-	$x^{3}$
					-	$x^{3}$	+	$0 x^{2}$
					+	$x^{3}$	+	$x^{2}$
							+	$x^{2}$	-	$8 x$
							-	$x^{2}$	-	$x$
									-	$9 x$	-	$9$
									-	$9 x$	-	$9$

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x - 9$

$x^{3}$

$x^{2}$

$x$

$9$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$8 x$

$x^{2}$

$x$

$9 x$

$9$

$9 x$

$9$

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

$x^{3}$

$x^{2}$

$x$

$9$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$8 x$

$9$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$8 x$

$x^{2}$

$x$

$9 x$

$9$

$9 x$

$9$

$0$

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

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

Write $x^{4} - 8 x - 9$ as a set of factors.

$x ((x + 1) (x^{3} - x^{2} + x - 9)) - 3 x^{4} - 8 x^{3} - 5 = 0$

Remove unnecessary parentheses.

$x (x + 1) (x^{3} - x^{2} + x - 9) - 3 x^{4} - 8 x^{3} - 5 = 0$

Factor $- 3 x^{4} - 8 x^{3} - 5$ 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.

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

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

$\pm 1, \pm 0.\overline{3}, \pm 5, \pm 1.\overline{6}$

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.

$- 3 {(- 1)}^{4} - 8 {(- 1)}^{3} - 5$

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

$- 3 \cdot 1 - 8 {(- 1)}^{3} - 5$

Multiply $- 3$ by $1$ .

$- 3 - 8 {(- 1)}^{3} - 5$

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

$- 3 - 8 \cdot - 1 - 5$

Multiply $- 8$ by $- 1$ .

$- 3 + 8 - 5$

Add $- 3$ and $8$ .

$5 - 5$

Subtract $5$ from $5$ .

$0$

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.

$\frac{- 3 x^{4} - 8 x^{3} - 5}{x + 1}$

Divide $- 3 x^{4} - 8 x^{3} - 5$ 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 $0$ .

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

Divide the highest order term in the dividend $- 3 x^{4}$ by the highest order term in divisor $x$ .

				-	$3 x^{3}$
	$x$	+	$1$	-	$3 x^{4}$	-	$8 x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$5$

Multiply the new quotient term by the divisor.

			-	$3 x^{3}$
$x$	+	$1$	-	$3 x^{4}$	-	$8 x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$5$
			-	$3 x^{4}$	-	$3 x^{3}$

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

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

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

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

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

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

Divide the highest order term in the dividend $- 5 x^{3}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$5 x^{2}$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$5 x^{2}$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{3} - 5 x^{2}$

$3 x^{3}$

$5 x^{2}$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

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

$3 x^{3}$

$5 x^{2}$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

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

$3 x^{3}$

$5 x^{2}$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$0 x$

Divide the highest order term in the dividend $5 x^{2}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$5 x^{2}$

$5 x$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$0 x$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$5 x^{2}$

$5 x$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} + 5 x$

$3 x^{3}$

$5 x^{2}$

$5 x$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

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

$3 x^{3}$

$5 x^{2}$

$5 x$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

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

$3 x^{3}$

$5 x^{2}$

$5 x$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

$5$

Divide the highest order term in the dividend $- 5 x$ by the highest order term in divisor $x$ .

$3 x^{3}$

$5 x^{2}$

$5 x$

$5$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

$5$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$5 x^{2}$

$5 x$

$5$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

$5$

$5 x$

$5$

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x - 5$

$3 x^{3}$

$5 x^{2}$

$5 x$

$5$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

$5$

$5 x$

$5$

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

$3 x^{3}$

$5 x^{2}$

$5 x$

$5$

$x$

$1$

$3 x^{4}$

$8 x^{3}$

$0 x^{2}$

$0 x$

$5$

$3 x^{4}$

$3 x^{3}$

$5 x^{3}$

$0 x^{2}$

$5 x^{3}$

$5 x^{2}$

$0 x$

$5 x^{2}$

$5 x$

$5$

$5 x$

$5$

$0$

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

$- 3 x^{3} - 5 x^{2} + 5 x - 5$

Write $- 3 x^{4} - 8 x^{3} - 5$ as a set of factors.

$x (x + 1) (x^{3} - x^{2} + x - 9) + (x + 1) (- 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

Factor $x + 1$ out of $x (x + 1) (x^{3} - x^{2} + x - 9) + (x + 1) (- 3 x^{3} - 5 x^{2} + 5 x - 5)$ .

Factor $x + 1$ out of $x (x + 1) (x^{3} - x^{2} + x - 9)$ .

$(x + 1) (x (x^{3} - x^{2} + x - 9)) + (x + 1) (- 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

Factor $x + 1$ out of $(x + 1) (x (x^{3} - x^{2} + x - 9)) + (x + 1) (- 3 x^{3} - 5 x^{2} + 5 x - 5)$ .

$(x + 1) (x (x^{3} - x^{2} + x - 9) - 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

Apply the distributive property.

$(x + 1) (x \cdot x^{3} + x (- x^{2}) + x \cdot x + x \cdot - 9 - 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

Simplify.

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

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

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

$(x + 1) (x \cdot x^{3} + x (- x^{2}) + x \cdot x + x \cdot - 9 - 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

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

$(x + 1) (x^{1 + 3} + x (- x^{2}) + x \cdot x + x \cdot - 9 - 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

Add $1$ and $3$ .

$(x + 1) (x^{4} + x (- x^{2}) + x \cdot x + x \cdot - 9 - 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

Rewrite using the commutative property of multiplication.

$(x + 1) (x^{4} - x \cdot x^{2} + x \cdot x + x \cdot - 9 - 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

Multiply $x$ by $x$ .

$(x + 1) (x^{4} - x \cdot x^{2} + x^{2} + x \cdot - 9 - 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

Move $- 9$ to the left of $x$ .

$(x + 1) (x^{4} - x \cdot x^{2} + x^{2} - 9 \cdot x - 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(x + 1) (x^{4} - (x^{2} x) + x^{2} - 9 \cdot x - 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

Multiply $x^{2}$ by $x$ .

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

$(x + 1) (x^{4} - (x^{2} x) + x^{2} - 9 \cdot x - 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

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

$(x + 1) (x^{4} - x^{2 + 1} + x^{2} - 9 \cdot x - 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

Add $2$ and $1$ .

$(x + 1) (x^{4} - x^{3} + x^{2} - 9 \cdot x - 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

$(x + 1) (x^{4} - x^{3} + x^{2} - 9 x - 3 x^{3} - 5 x^{2} + 5 x - 5) = 0$

Subtract $3 x^{3}$ from $- x^{3}$ .

$(x + 1) (x^{4} - 4 x^{3} + x^{2} - 9 x - 5 x^{2} + 5 x - 5) = 0$

Subtract $5 x^{2}$ from $x^{2}$ .

$(x + 1) (x^{4} - 4 x^{3} - 4 x^{2} - 9 x + 5 x - 5) = 0$

Add $- 9 x$ and $5 x$ .

$(x + 1) (x^{4} - 4 x^{3} - 4 x^{2} - 4 x - 5) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$x^{4} - 4 x^{3} - 4 x^{2} - 4 x - 5 = 0$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $x^{4} - 4 x^{3} - 4 x^{2} - 4 x - 5$ equal to $0$ and solve for $x$ .

Set $x^{4} - 4 x^{3} - 4 x^{2} - 4 x - 5$ equal to $0$ .

$x^{4} - 4 x^{3} - 4 x^{2} - 4 x - 5 = 0$

Solve $x^{4} - 4 x^{3} - 4 x^{2} - 4 x - 5 = 0$ for $x$ .

Factor the left side of the equation.

Regroup terms.

$- 4 x^{3} - 4 x^{2} + x^{4} - 4 x - 5 = 0$

Factor $- 4 x^{2}$ out of $- 4 x^{3} - 4 x^{2}$ .

Factor $- 4 x^{2}$ out of $- 4 x^{3}$ .

$- 4 x^{2} x - 4 x^{2} + x^{4} - 4 x - 5 = 0$

Factor $- 4 x^{2}$ out of $- 4 x^{2}$ .

$- 4 x^{2} x - 4 x^{2} \cdot 1 + x^{4} - 4 x - 5 = 0$

Factor $- 4 x^{2}$ out of $- 4 x^{2} (x) - 4 x^{2} (1)$ .

$- 4 x^{2} (x + 1) + x^{4} - 4 x - 5 = 0$

Factor $x^{4} - 4 x - 5$ 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.

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

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

$\pm 1, \pm 5$

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.

${(- 1)}^{4} - 4 \cdot - 1 - 5$

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

$1 - 4 \cdot - 1 - 5$

Multiply $- 4$ by $- 1$ .

$1 + 4 - 5$

Add $1$ and $4$ .

$5 - 5$

Subtract $5$ from $5$ .

$0$

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.

$\frac{x^{4} - 4 x - 5}{x + 1}$

Divide $x^{4} - 4 x - 5$ 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 $0$ .

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

$0 x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$4 x$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$4 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$4 x$

$x^{2}$

$x$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$4 x$

$x^{2}$

$x$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$4 x$

$x^{2}$

$x$

$5 x$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$4 x$

$x^{2}$

$x$

$5 x$

$5$

Divide the highest order term in the dividend $- 5 x$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2}$

$x$

$5$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$4 x$

$x^{2}$

$x$

$5 x$

$5$

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$x^{2}$	+	$x$	-	$5$
$x$	+	$1$		$x^{4}$	+	$0 x^{3}$	+	$0 x^{2}$	-	$4 x$	-	$5$
			-	$x^{4}$	-	$x^{3}$
					-	$x^{3}$	+	$0 x^{2}$
					+	$x^{3}$	+	$x^{2}$
							+	$x^{2}$	-	$4 x$
							-	$x^{2}$	-	$x$
									-	$5 x$	-	$5$
									-	$5 x$	-	$5$

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x - 5$

$x^{3}$

$x^{2}$

$x$

$5$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$4 x$

$x^{2}$

$x$

$5 x$

$5$

$5 x$

$5$

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

$x^{3}$

$x^{2}$

$x$

$5$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$4 x$

$5$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$4 x$

$x^{2}$

$x$

$5 x$

$5$

$5 x$

$5$

$0$

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

$x^{3} - x^{2} + x - 5$

Write $x^{4} - 4 x - 5$ as a set of factors.

$- 4 x^{2} (x + 1) + (x + 1) (x^{3} - x^{2} + x - 5) = 0$

Factor $x + 1$ out of $- 4 x^{2} (x + 1) + (x + 1) (x^{3} - x^{2} + x - 5)$ .

Factor $x + 1$ out of $- 4 x^{2} (x + 1)$ .

$(x + 1) (- 4 x^{2}) + (x + 1) (x^{3} - x^{2} + x - 5) = 0$

Factor $x + 1$ out of $(x + 1) (- 4 x^{2}) + (x + 1) (x^{3} - x^{2} + x - 5)$ .

$(x + 1) (- 4 x^{2} + x^{3} - x^{2} + x - 5) = 0$

Subtract $x^{2}$ from $- 4 x^{2}$ .

$(x + 1) (x^{3} - 5 x^{2} + x - 5) = 0$

Factor.

Rewrite $x^{3} - 5 x^{2} + x - 5$ in a factored form.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x + 1) ((x^{3} - 5 x^{2}) + x - 5) = 0$

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

$(x + 1) (x^{2} (x - 5) + 1 (x - 5)) = 0$

Factor the polynomial by factoring out the greatest common factor, $x - 5$ .

$(x + 1) ((x - 5) (x^{2} + 1)) = 0$

Remove unnecessary parentheses.

$(x + 1) (x - 5) (x^{2} + 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$x - 5 = 0$

$x^{2} + 1 = 0$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Solve $x^{2} + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

The final solution is all the values that make $(x + 1) (x - 5) (x^{2} + 1) = 0$ true.

$x = - 1, 5, i, - i$

The final solution is all the values that make $(x + 1) (x^{4} - 4 x^{3} - 4 x^{2} - 4 x - 5) = 0$ true.

$x = - 1, 5, i, - i$

Find the Roots (Zeros) x^5-3x^4-8x^3-8x^2-9x-5=0

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