Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3-13x^2+59x-87

$f (x) = x^{3} - 13 x^{2} + 59 x - 87$

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.

$p = \pm 1, \pm 3, \pm 29, \pm 87$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 3, \pm 29, \pm 87$

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.

${(3)}^{3} - 13 {(3)}^{2} + 59 (3) - 87$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 3$ is a root of the polynomial.

Simplify each term.

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

$27 - 13 {(3)}^{2} + 59 (3) - 87$

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

$27 - 13 \cdot 9 + 59 (3) - 87$

Multiply $- 13$ by $9$ .

$27 - 117 + 59 (3) - 87$

Multiply $59$ by $3$ .

$27 - 117 + 177 - 87$

Simplify by adding and subtracting.

Subtract $117$ from $27$ .

$- 90 + 177 - 87$

Add $- 90$ and $177$ .

$87 - 87$

Subtract $87$ from $87$ .

$0$

Step 5

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.

$\frac{x^{3} - 13 x^{2} + 59 x - 87}{x - 3}$

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.

$3$	$1$	$- 13$	$59$	$- 87$
	\xa0			\xa0

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

$3$	$1$	$- 13$	$59$	$- 87$
	\xa0			\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(3)$ and place the result of $(3)$ under the next term in the dividend $(- 13)$ .

$3$	$1$	$- 13$	$59$	$- 87$
	\xa0	$3$		\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.

$3$	$1$	$- 13$	$59$	$- 87$
	\xa0	$3$		\xa0
	$1$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(3)$ and place the result of $(- 30)$ under the next term in the dividend $(59)$ .

$3$	$1$	$- 13$	$59$	$- 87$
	\xa0	$3$	$- 30$	\xa0
	$1$	$- 10$

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

$3$	$1$	$- 13$	$59$	$- 87$
	\xa0	$3$	$- 30$	\xa0
	$1$	$- 10$	$29$

Multiply the newest entry in the result $(29)$ by the divisor $(3)$ and place the result of $(87)$ under the next term in the dividend $(- 87)$ .

$3$	$1$	$- 13$	$59$	$- 87$
	\xa0	$3$	$- 30$	$87$
	$1$	$- 10$	$29$

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

$3$	$1$	$- 13$	$59$	$- 87$
	\xa0	$3$	$- 30$	$87$
	$1$	$- 10$	$29$	$0$

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

$1 x^{2} + - 10 x + 29$

Simplify the quotient polynomial.

$x^{2} - 10 x + 29$

Step 7

Solve the equation to find any remaining roots.

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 10$ , and $c = 29$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot 29)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 29$ .

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 29}}{2 \cdot 1}$

Multiply $- 4$ by $29$ .

$x = \frac{10 \pm \sqrt{100 - 116}}{2 \cdot 1}$

Subtract $116$ from $100$ .

$x = \frac{10 \pm \sqrt{- 16}}{2 \cdot 1}$

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

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

$x = \frac{10 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

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

$x = \frac{10 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

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

$x = \frac{10 \pm i \cdot 4}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{10 \pm 4 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{10 \pm 4 i}{2}$

Simplify $\frac{10 \pm 4 i}{2}$ .

$x = 5 \pm 2 i$

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

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 29$ .

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 29}}{2 \cdot 1}$

Multiply $- 4$ by $29$ .

$x = \frac{10 \pm \sqrt{100 - 116}}{2 \cdot 1}$

Subtract $116$ from $100$ .

$x = \frac{10 \pm \sqrt{- 16}}{2 \cdot 1}$

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

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

$x = \frac{10 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

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

$x = \frac{10 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

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

$x = \frac{10 \pm i \cdot 4}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{10 \pm 4 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{10 \pm 4 i}{2}$

Simplify $\frac{10 \pm 4 i}{2}$ .

$x = 5 \pm 2 i$

Change the $\pm$ to $+$ .

$x = 5 + 2 i$

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

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 29$ .

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 29}}{2 \cdot 1}$

Multiply $- 4$ by $29$ .

$x = \frac{10 \pm \sqrt{100 - 116}}{2 \cdot 1}$

Subtract $116$ from $100$ .

$x = \frac{10 \pm \sqrt{- 16}}{2 \cdot 1}$

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

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

$x = \frac{10 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

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

$x = \frac{10 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

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

$x = \frac{10 \pm i \cdot 4}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{10 \pm 4 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{10 \pm 4 i}{2}$

Simplify $\frac{10 \pm 4 i}{2}$ .

$x = 5 \pm 2 i$

Change the $\pm$ to $-$ .

$x = 5 - 2 i$

The final answer is the combination of both solutions.

$x = 5 + 2 i, 5 - 2 i$

Step 8

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

$(x - 3) (x - 5 - 2 i) (x - 5 + 2 i)$

Step 9

These are the roots (zeros) of the polynomial $x^{3} - 13 x^{2} + 59 x - 87$ .

$x = 3, 5 + 2 i, 5 - 2 i$

Step 10

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3-13x^2+59x-87

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