Find the Roots (Zeros) 4x^4+31x^3-4x^2-89x+22=0

$4 x^{4} + 31 x^{3} - 4 x^{2} - 89 x + 22 = 0$

Step 1

Factor the left side of the equation.

Factor $4 x^{4} + 31 x^{3} - 4 x^{2} - 89 x + 22$ 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.

$p = \pm 1, \pm 22, \pm 2, \pm 11 q = \pm 1, \pm 4, \pm 2$

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 22, \pm 5.5, \pm 11, \pm 2, \pm 2.75$

Substitute $0.25$ and simplify the expression. In this case, the expression is equal to $0$ so $0.25$ is a root of the polynomial.

Substitute $0.25$ into the polynomial.

$4 \cdot {0.25}^{4} + 31 \cdot {0.25}^{3} - 4 \cdot {0.25}^{2} - 89 \cdot 0.25 + 22$

Raise $0.25$ to the power of $4$ .

$4 \cdot 0.00390625 + 31 \cdot {0.25}^{3} - 4 \cdot {0.25}^{2} - 89 \cdot 0.25 + 22$

Multiply $4$ by $0.00390625$ .

$0.015625 + 31 \cdot {0.25}^{3} - 4 \cdot {0.25}^{2} - 89 \cdot 0.25 + 22$

Raise $0.25$ to the power of $3$ .

$0.015625 + 31 \cdot 0.015625 - 4 \cdot {0.25}^{2} - 89 \cdot 0.25 + 22$

Multiply $31$ by $0.015625$ .

$0.015625 + 0.484375 - 4 \cdot {0.25}^{2} - 89 \cdot 0.25 + 22$

Add $0.015625$ and $0.484375$ .

$0.5 - 4 \cdot {0.25}^{2} - 89 \cdot 0.25 + 22$

Raise $0.25$ to the power of $2$ .

$0.5 - 4 \cdot 0.0625 - 89 \cdot 0.25 + 22$

Multiply $- 4$ by $0.0625$ .

$0.5 - 0.25 - 89 \cdot 0.25 + 22$

Subtract $0.25$ from $0.5$ .

$0.25 - 89 \cdot 0.25 + 22$

Multiply $- 89$ by $0.25$ .

$0.25 - 22.25 + 22$

Subtract $22.25$ from $0.25$ .

$- 22 + 22$

Add $- 22$ and $22$ .

$0$

Since $0.25$ is a known root, divide the polynomial by $4 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{4} + 31 x^{3} - 4 x^{2} - 89 x + 22}{4 x - 1}$

Divide $4 x^{4} + 31 x^{3} - 4 x^{2} - 89 x + 22$ by $4 x - 1$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

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

$x^{3}$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

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

$x^{3}$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

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

$x^{3}$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

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

$x^{3}$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

Divide the highest order term in the dividend $32 x^{3}$ by the highest order term in divisor $4 x$ .

$x^{3}$

$8 x^{2}$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$8 x^{2}$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

$32 x^{3}$

$8 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $32 x^{3} - 8 x^{2}$

$x^{3}$

$8 x^{2}$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

$32 x^{3}$

$8 x^{2}$

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

$x^{3}$

$8 x^{2}$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

$32 x^{3}$

$8 x^{2}$

$4 x^{2}$

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

$x^{3}$

$8 x^{2}$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

$32 x^{3}$

$8 x^{2}$

$4 x^{2}$

$89 x$

Divide the highest order term in the dividend $4 x^{2}$ by the highest order term in divisor $4 x$ .

$x^{3}$

$8 x^{2}$

$x$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

$32 x^{3}$

$8 x^{2}$

$4 x^{2}$

$89 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$8 x^{2}$

$x$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

$32 x^{3}$

$8 x^{2}$

$4 x^{2}$

$89 x$

$4 x^{2}$

$x$

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{2} - x$

$x^{3}$

$8 x^{2}$

$x$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

$32 x^{3}$

$8 x^{2}$

$4 x^{2}$

$89 x$

$4 x^{2}$

$x$

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

$x^{3}$

$8 x^{2}$

$x$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

$32 x^{3}$

$8 x^{2}$

$4 x^{2}$

$89 x$

$4 x^{2}$

$x$

$88 x$

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

$x^{3}$

$8 x^{2}$

$x$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

$32 x^{3}$

$8 x^{2}$

$4 x^{2}$

$89 x$

$4 x^{2}$

$x$

$88 x$

$22$

Divide the highest order term in the dividend $- 88 x$ by the highest order term in divisor $4 x$ .

$x^{3}$

$8 x^{2}$

$x$

$22$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

$32 x^{3}$

$8 x^{2}$

$4 x^{2}$

$89 x$

$4 x^{2}$

$x$

$88 x$

$22$

Multiply the new quotient term by the divisor.

$x^{3}$

$8 x^{2}$

$x$

$22$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

$32 x^{3}$

$8 x^{2}$

$4 x^{2}$

$89 x$

$4 x^{2}$

$x$

$88 x$

$22$

$88 x$

$22$

The expression needs to be subtracted from the dividend, so change all the signs in $- 88 x + 22$

$x^{3}$

$8 x^{2}$

$x$

$22$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

$32 x^{3}$

$8 x^{2}$

$4 x^{2}$

$89 x$

$4 x^{2}$

$x$

$88 x$

$22$

$88 x$

$22$

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

$x^{3}$

$8 x^{2}$

$x$

$22$

$4 x$

$1$

$4 x^{4}$

$31 x^{3}$

$4 x^{2}$

$89 x$

$22$

$4 x^{4}$

$x^{3}$

$32 x^{3}$

$4 x^{2}$

$32 x^{3}$

$8 x^{2}$

$4 x^{2}$

$89 x$

$4 x^{2}$

$x$

$88 x$

$22$

$88 x$

$22$

$0$

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

$x^{3} + 8 x^{2} + x - 22$

Write $4 x^{4} + 31 x^{3} - 4 x^{2} - 89 x + 22$ as a set of factors.

$(4 x - 1) (x^{3} + 8 x^{2} + x - 22) = 0$

Factor $x^{3} + 8 x^{2} + x - 22$ 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.

$p = \pm 1, \pm 22, \pm 2, \pm 11 q = \pm 1$

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

$\pm 1, \pm 22, \pm 2, \pm 11$

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

Substitute $- 2$ into the polynomial.

${(- 2)}^{3} + 8 {(- 2)}^{2} - 2 - 22$

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

$- 8 + 8 {(- 2)}^{2} - 2 - 22$

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

$- 8 + 8 \cdot 4 - 2 - 22$

Multiply $8$ by $4$ .

$- 8 + 32 - 2 - 22$

Add $- 8$ and $32$ .

$24 - 2 - 22$

Subtract $2$ from $24$ .

$22 - 22$

Subtract $22$ from $22$ .

$0$

Since $- 2$ is a known root, divide the polynomial by $x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 8 x^{2} + x - 22}{x + 2}$

Divide $x^{3} + 8 x^{2} + x - 22$ by $x + 2$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$x^{3}$

$8 x^{2}$

$x$

$22$

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	+	$8 x^{2}$	+	$x$	-	$22$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$8 x^{2}$	+	$x$	-	$22$
			+	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$8 x^{2}$	+	$x$	-	$22$
			-	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$8 x^{2}$	+	$x$	-	$22$
			-	$x^{3}$	-	$2 x^{2}$
					+	$6 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$8 x^{2}$	+	$x$	-	$22$
			-	$x^{3}$	-	$2 x^{2}$
					+	$6 x^{2}$	+	$x$

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$6 x$
$x$	+	$2$		$x^{3}$	+	$8 x^{2}$	+	$x$	-	$22$
			-	$x^{3}$	-	$2 x^{2}$
					+	$6 x^{2}$	+	$x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$
$x$	+	$2$		$x^{3}$	+	$8 x^{2}$	+	$x$	-	$22$
			-	$x^{3}$	-	$2 x^{2}$
					+	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$12 x$

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} + 12 x$

				$x^{2}$	+	$6 x$
$x$	+	$2$		$x^{3}$	+	$8 x^{2}$	+	$x$	-	$22$
			-	$x^{3}$	-	$2 x^{2}$
					+	$6 x^{2}$	+	$x$
					-	$6 x^{2}$	-	$12 x$

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

				$x^{2}$	+	$6 x$
$x$	+	$2$		$x^{3}$	+	$8 x^{2}$	+	$x$	-	$22$
			-	$x^{3}$	-	$2 x^{2}$
					+	$6 x^{2}$	+	$x$
					-	$6 x^{2}$	-	$12 x$
							-	$11 x$

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

				$x^{2}$	+	$6 x$
$x$	+	$2$		$x^{3}$	+	$8 x^{2}$	+	$x$	-	$22$
			-	$x^{3}$	-	$2 x^{2}$
					+	$6 x^{2}$	+	$x$
					-	$6 x^{2}$	-	$12 x$
							-	$11 x$	-	$22$

Divide the highest order term in the dividend $- 11 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$6 x$	-	$11$
$x$	+	$2$		$x^{3}$	+	$8 x^{2}$	+	$x$	-	$22$
			-	$x^{3}$	-	$2 x^{2}$
					+	$6 x^{2}$	+	$x$
					-	$6 x^{2}$	-	$12 x$
							-	$11 x$	-	$22$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$	-	$11$
$x$	+	$2$		$x^{3}$	+	$8 x^{2}$	+	$x$	-	$22$
			-	$x^{3}$	-	$2 x^{2}$
					+	$6 x^{2}$	+	$x$
					-	$6 x^{2}$	-	$12 x$
							-	$11 x$	-	$22$
							-	$11 x$	-	$22$

The expression needs to be subtracted from the dividend, so change all the signs in $- 11 x - 22$

				$x^{2}$	+	$6 x$	-	$11$
$x$	+	$2$		$x^{3}$	+	$8 x^{2}$	+	$x$	-	$22$
			-	$x^{3}$	-	$2 x^{2}$
					+	$6 x^{2}$	+	$x$
					-	$6 x^{2}$	-	$12 x$
							-	$11 x$	-	$22$
							+	$11 x$	+	$22$

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

				$x^{2}$	+	$6 x$	-	$11$
$x$	+	$2$		$x^{3}$	+	$8 x^{2}$	+	$x$	-	$22$
			-	$x^{3}$	-	$2 x^{2}$
					+	$6 x^{2}$	+	$x$
					-	$6 x^{2}$	-	$12 x$
							-	$11 x$	-	$22$
							+	$11 x$	+	$22$
										$0$

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

$x^{2} + 6 x - 11$

Write $x^{3} + 8 x^{2} + x - 22$ as a set of factors.

$(4 x - 1) ((x + 2) (x^{2} + 6 x - 11)) = 0$

Remove unnecessary parentheses.

$(4 x - 1) (x + 2) (x^{2} + 6 x - 11) = 0$

Step 2

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

$4 x - 1 = 0$

$x + 2 = 0$

$x^{2} + 6 x - 11 = 0$

Step 3

Set $4 x - 1$ equal to $0$ and solve for $x$ .

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Solve $4 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$4 x = 1$

Divide each term in $4 x = 1$ by $4$ and simplify.

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 4

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

Set $x^{2} + 6 x - 11$ equal to $0$ and solve for $x$ .

Set $x^{2} + 6 x - 11$ equal to $0$ .

$x^{2} + 6 x - 11 = 0$

Solve $x^{2} + 6 x - 11 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 6$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot - 11)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 11$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4$ by $- 11$ .

$x = \frac{- 6 \pm \sqrt{36 + 44}}{2 \cdot 1}$

Add $36$ and $44$ .

$x = \frac{- 6 \pm \sqrt{80}}{2 \cdot 1}$

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

$x = \frac{- 6 \pm \sqrt{16 (5)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 6 \pm \sqrt{4^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 6 \pm 4 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 4 \sqrt{5}}{2}$

Simplify $\frac{- 6 \pm 4 \sqrt{5}}{2}$ .

$x = - 3 \pm 2 \sqrt{5}$

The final answer is the combination of both solutions.

$x = - 3 + 2 \sqrt{5}, - 3 - 2 \sqrt{5}$

Step 6

The final solution is all the values that make $(4 x - 1) (x + 2) (x^{2} + 6 x - 11) = 0$ true.

$x = \frac{1}{4}, - 2, - 3 + 2 \sqrt{5}, - 3 - 2 \sqrt{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{4}, - 2, - 3 + 2 \sqrt{5}, - 3 - 2 \sqrt{5}$

Decimal Form:

$x = 0.25, - 2, 1.47213595 \dots, - 7.47213595 \dots$

Step 8

Find the Roots (Zeros) 4x^4+31x^3-4x^2-89x+22=0

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