Find the Roots/Zeros Using the Rational Roots Test f(x)=x^4-8x^3+x^2+128x-272

$f (x) = x^{4} - 8 x^{3} + x^{2} + 128 x - 272$

Step 1

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.

$p = \pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm 17, \pm 34, \pm 68, \pm 136, \pm 272$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm 17, \pm 34, \pm 68, \pm 136, \pm 272$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(4)}^{4} - 8 {(4)}^{3} + {(4)}^{2} + 128 (4) - 272$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 4$ is a root of the polynomial.

Simplify each term.

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

$256 - 8 {(4)}^{3} + {(4)}^{2} + 128 (4) - 272$

Raise $4$ to the power of $3$ .

$256 - 8 \cdot 64 + {(4)}^{2} + 128 (4) - 272$

Multiply $- 8$ by $64$ .

$256 - 512 + {(4)}^{2} + 128 (4) - 272$

Raise $4$ to the power of $2$ .

$256 - 512 + 16 + 128 (4) - 272$

Multiply $128$ by $4$ .

$256 - 512 + 16 + 512 - 272$

Simplify by adding and subtracting.

Subtract $512$ from $256$ .

$- 256 + 16 + 512 - 272$

Add $- 256$ and $16$ .

$- 240 + 512 - 272$

Add $- 240$ and $512$ .

$272 - 272$

Subtract $272$ from $272$ .

$0$

Step 5

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

$\frac{x^{4} - 8 x^{3} + x^{2} + 128 x - 272}{x - 4}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$4$	$1$	$- 8$	$1$	$128$	$- 272$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$4$	$1$	$- 8$	$1$	$128$	$- 272$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(4)$ and place the result of $(4)$ under the next term in the dividend $(- 8)$ .

$4$	$1$	$- 8$	$1$	$128$	$- 272$
	\xa0	$4$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$4$	$1$	$- 8$	$1$	$128$	$- 272$
	\xa0	$4$			\xa0
	$1$	$- 4$

Multiply the newest entry in the result $(- 4)$ by the divisor $(4)$ and place the result of $(- 16)$ under the next term in the dividend $(1)$ .

$4$	$1$	$- 8$	$1$	$128$	$- 272$
	\xa0	$4$	$- 16$		\xa0
	$1$	$- 4$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$4$	$1$	$- 8$	$1$	$128$	$- 272$
	\xa0	$4$	$- 16$		\xa0
	$1$	$- 4$	$- 15$

Multiply the newest entry in the result $(- 15)$ by the divisor $(4)$ and place the result of $(- 60)$ under the next term in the dividend $(128)$ .

$4$	$1$	$- 8$	$1$	$128$	$- 272$
	\xa0	$4$	$- 16$	$- 60$	\xa0
	$1$	$- 4$	$- 15$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$4$	$1$	$- 8$	$1$	$128$	$- 272$
	\xa0	$4$	$- 16$	$- 60$	\xa0
	$1$	$- 4$	$- 15$	$68$

Multiply the newest entry in the result $(68)$ by the divisor $(4)$ and place the result of $(272)$ under the next term in the dividend $(- 272)$ .

$4$	$1$	$- 8$	$1$	$128$	$- 272$
	\xa0	$4$	$- 16$	$- 60$	$272$
	$1$	$- 4$	$- 15$	$68$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$4$	$1$	$- 8$	$1$	$128$	$- 272$
	\xa0	$4$	$- 16$	$- 60$	$272$
	$1$	$- 4$	$- 15$	$68$	$0$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + - 4 x^{2} + (- 15) x + 68$

Simplify the quotient polynomial.

$x^{3} - 4 x^{2} - 15 x + 68$

Step 7

Solve the equation to find any remaining roots.

Factor $x^{3} - 4 x^{2} - 15 x + 68$ 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.

$p = \pm 1, \pm 68, \pm 2, \pm 34, \pm 4, \pm 17 q = \pm 1$

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

$\pm 1, \pm 68, \pm 2, \pm 34, \pm 4, \pm 17$

Substitute $- 4$ and simplify the expression. In this case, the expression is equal to $0$ so $- 4$ is a root of the polynomial.

Substitute $- 4$ into the polynomial.

${(- 4)}^{3} - 4 {(- 4)}^{2} - 15 \cdot - 4 + 68$

Raise $- 4$ to the power of $3$ .

$- 64 - 4 {(- 4)}^{2} - 15 \cdot - 4 + 68$

Raise $- 4$ to the power of $2$ .

$- 64 - 4 \cdot 16 - 15 \cdot - 4 + 68$

Multiply $- 4$ by $16$ .

$- 64 - 64 - 15 \cdot - 4 + 68$

Subtract $64$ from $- 64$ .

$- 128 - 15 \cdot - 4 + 68$

Multiply $- 15$ by $- 4$ .

$- 128 + 60 + 68$

Add $- 128$ and $60$ .

$- 68 + 68$

Add $- 68$ and $68$ .

$0$

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

$\frac{x^{3} - 4 x^{2} - 15 x + 68}{x + 4}$

Divide $x^{3} - 4 x^{2} - 15 x + 68$ by $x + 4$ .

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

$x$

$4$

$x^{3}$

$4 x^{2}$

$15 x$

$68$

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

					$x^{2}$
	$x$	+	$4$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$68$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$4$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$68$
			+	$x^{3}$	+	$4 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 4 x^{2}$

				$x^{2}$
$x$	+	$4$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$68$
			-	$x^{3}$	-	$4 x^{2}$

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

				$x^{2}$
$x$	+	$4$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$68$
			-	$x^{3}$	-	$4 x^{2}$
					-	$8 x^{2}$

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

				$x^{2}$
$x$	+	$4$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$68$
			-	$x^{3}$	-	$4 x^{2}$
					-	$8 x^{2}$	-	$15 x$

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

				$x^{2}$	-	$8 x$
$x$	+	$4$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$68$
			-	$x^{3}$	-	$4 x^{2}$
					-	$8 x^{2}$	-	$15 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$
$x$	+	$4$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$68$
			-	$x^{3}$	-	$4 x^{2}$
					-	$8 x^{2}$	-	$15 x$
					-	$8 x^{2}$	-	$32 x$

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

				$x^{2}$	-	$8 x$
$x$	+	$4$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$68$
			-	$x^{3}$	-	$4 x^{2}$
					-	$8 x^{2}$	-	$15 x$
					+	$8 x^{2}$	+	$32 x$

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

				$x^{2}$	-	$8 x$
$x$	+	$4$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$68$
			-	$x^{3}$	-	$4 x^{2}$
					-	$8 x^{2}$	-	$15 x$
					+	$8 x^{2}$	+	$32 x$
							+	$17 x$

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

				$x^{2}$	-	$8 x$
$x$	+	$4$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$68$
			-	$x^{3}$	-	$4 x^{2}$
					-	$8 x^{2}$	-	$15 x$
					+	$8 x^{2}$	+	$32 x$
							+	$17 x$	+	$68$

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

				$x^{2}$	-	$8 x$	+	$17$
$x$	+	$4$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$68$
			-	$x^{3}$	-	$4 x^{2}$
					-	$8 x^{2}$	-	$15 x$
					+	$8 x^{2}$	+	$32 x$
							+	$17 x$	+	$68$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$	+	$17$
$x$	+	$4$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$68$
			-	$x^{3}$	-	$4 x^{2}$
					-	$8 x^{2}$	-	$15 x$
					+	$8 x^{2}$	+	$32 x$
							+	$17 x$	+	$68$
							+	$17 x$	+	$68$

The expression needs to be subtracted from the dividend, so change all the signs in $17 x + 68$

				$x^{2}$	-	$8 x$	+	$17$
$x$	+	$4$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$68$
			-	$x^{3}$	-	$4 x^{2}$
					-	$8 x^{2}$	-	$15 x$
					+	$8 x^{2}$	+	$32 x$
							+	$17 x$	+	$68$
							-	$17 x$	-	$68$

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

				$x^{2}$	-	$8 x$	+	$17$
$x$	+	$4$		$x^{3}$	-	$4 x^{2}$	-	$15 x$	+	$68$
			-	$x^{3}$	-	$4 x^{2}$
					-	$8 x^{2}$	-	$15 x$
					+	$8 x^{2}$	+	$32 x$
							+	$17 x$	+	$68$
							-	$17 x$	-	$68$
										$0$

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

$x^{2} - 8 x + 17$

Write $x^{3} - 4 x^{2} - 15 x + 68$ as a set of factors.

$(x + 4) (x^{2} - 8 x + 17) = 0$

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

$x + 4 = 0$

$x^{2} - 8 x + 17 = 0$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Set $x^{2} - 8 x + 17$ equal to $0$ and solve for $x$ .

Set $x^{2} - 8 x + 17$ equal to $0$ .

$x^{2} - 8 x + 17 = 0$

Solve $x^{2} - 8 x + 17 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 8$ , and $c = 17$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 17)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 17}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 17$ .

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 17}}{2 \cdot 1}$

Multiply $- 4$ by $17$ .

$x = \frac{8 \pm \sqrt{64 - 68}}{2 \cdot 1}$

Subtract $68$ from $64$ .

$x = \frac{8 \pm \sqrt{- 4}}{2 \cdot 1}$

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{8 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{8 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{8 \pm i \cdot 2}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{8 \pm 2 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 i}{2}$

Simplify $\frac{8 \pm 2 i}{2}$ .

$x = 4 \pm i$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 17}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 17$ .

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 17}}{2 \cdot 1}$

Multiply $- 4$ by $17$ .

$x = \frac{8 \pm \sqrt{64 - 68}}{2 \cdot 1}$

Subtract $68$ from $64$ .

$x = \frac{8 \pm \sqrt{- 4}}{2 \cdot 1}$

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{8 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{8 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{8 \pm i \cdot 2}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{8 \pm 2 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 i}{2}$

Simplify $\frac{8 \pm 2 i}{2}$ .

$x = 4 \pm i$

Change the $\pm$ to $+$ .

$x = 4 + i$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 17}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 17$ .

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 17}}{2 \cdot 1}$

Multiply $- 4$ by $17$ .

$x = \frac{8 \pm \sqrt{64 - 68}}{2 \cdot 1}$

Subtract $68$ from $64$ .

$x = \frac{8 \pm \sqrt{- 4}}{2 \cdot 1}$

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{8 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{8 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{8 \pm i \cdot 2}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{8 \pm 2 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 i}{2}$

Simplify $\frac{8 \pm 2 i}{2}$ .

$x = 4 \pm i$

Change the $\pm$ to $-$ .

$x = 4 - i$

The final answer is the combination of both solutions.

$x = 4 + i, 4 - i$

The final solution is all the values that make $(x + 4) (x^{2} - 8 x + 17) = 0$ true.

$x = - 4, 4 + i, 4 - i$

Step 8

The polynomial can be written as a set of linear factors.

$(x - 4) (x + 4) (x - 4 - i) (x - 4 + i)$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 8 x^{3} + x^{2} + 128 x - 272$ .

$x = 4, - 4, 4 + i, 4 - i$

Step 10

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^4-8x^3+x^2+128x-272

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