Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3-15x^2+73x-111

$f (x) = x^{3} - 15 x^{2} + 73 x - 111$

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.

$p = \pm 1, \pm 3, \pm 37, \pm 111$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 3, \pm 37, \pm 111$

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.

${(3)}^{3} - 15 {(3)}^{2} + 73 (3) - 111$

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 $3$ .

$27 - 15 {(3)}^{2} + 73 (3) - 111$

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

$27 - 15 \cdot 9 + 73 (3) - 111$

Multiply $- 15$ by $9$ .

$27 - 135 + 73 (3) - 111$

Multiply $73$ by $3$ .

$27 - 135 + 219 - 111$

Simplify by adding and subtracting.

Subtract $135$ from $27$ .

$- 108 + 219 - 111$

Add $- 108$ and $219$ .

$111 - 111$

Subtract $111$ from $111$ .

$0$

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.

$\frac{x^{3} - 15 x^{2} + 73 x - 111}{x - 3}$

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.

$3$	$1$	$- 15$	$73$	$- 111$
	\xa0			\xa0

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

$3$	$1$	$- 15$	$73$	$- 111$
	\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 $(- 15)$ .

$3$	$1$	$- 15$	$73$	$- 111$
	\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.

$3$	$1$	$- 15$	$73$	$- 111$
	\xa0	$3$		\xa0
	$1$	$- 12$

Multiply the newest entry in the result $(- 12)$ by the divisor $(3)$ and place the result of $(- 36)$ under the next term in the dividend $(73)$ .

$3$	$1$	$- 15$	$73$	$- 111$
	\xa0	$3$	$- 36$	\xa0
	$1$	$- 12$

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

$3$	$1$	$- 15$	$73$	$- 111$
	\xa0	$3$	$- 36$	\xa0
	$1$	$- 12$	$37$

Multiply the newest entry in the result $(37)$ by the divisor $(3)$ and place the result of $(111)$ under the next term in the dividend $(- 111)$ .

$3$	$1$	$- 15$	$73$	$- 111$
	\xa0	$3$	$- 36$	$111$
	$1$	$- 12$	$37$

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

$3$	$1$	$- 15$	$73$	$- 111$
	\xa0	$3$	$- 36$	$111$
	$1$	$- 12$	$37$	$0$

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

$1 x^{2} + - 12 x + 37$

Simplify the quotient polynomial.

$x^{2} - 12 x + 37$

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 $a = 1$ , $b = - 12$ , and $c = 37$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (1 \cdot 37)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 37}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 37$ .

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 37}}{2 \cdot 1}$

Multiply $- 4$ by $37$ .

$x = \frac{12 \pm \sqrt{144 - 148}}{2 \cdot 1}$

Subtract $148$ from $144$ .

$x = \frac{12 \pm \sqrt{- 4}}{2 \cdot 1}$

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

$x = \frac{12 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

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

$x = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{12 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{12 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

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

$x = \frac{12 \pm i \cdot 2}{2 \cdot 1}$

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

$x = \frac{12 \pm 2 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 i}{2}$

Simplify $\frac{12 \pm 2 i}{2}$ .

$x = 6 \pm i$

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

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 37}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 37$ .

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 37}}{2 \cdot 1}$

Multiply $- 4$ by $37$ .

$x = \frac{12 \pm \sqrt{144 - 148}}{2 \cdot 1}$

Subtract $148$ from $144$ .

$x = \frac{12 \pm \sqrt{- 4}}{2 \cdot 1}$

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

$x = \frac{12 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

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

$x = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{12 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{12 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

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

$x = \frac{12 \pm i \cdot 2}{2 \cdot 1}$

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

$x = \frac{12 \pm 2 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 i}{2}$

Simplify $\frac{12 \pm 2 i}{2}$ .

$x = 6 \pm i$

Change the $\pm$ to $+$ .

$x = 6 + i$

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

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 37}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 37$ .

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 37}}{2 \cdot 1}$

Multiply $- 4$ by $37$ .

$x = \frac{12 \pm \sqrt{144 - 148}}{2 \cdot 1}$

Subtract $148$ from $144$ .

$x = \frac{12 \pm \sqrt{- 4}}{2 \cdot 1}$

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

$x = \frac{12 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

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

$x = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{12 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{12 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

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

$x = \frac{12 \pm i \cdot 2}{2 \cdot 1}$

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

$x = \frac{12 \pm 2 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 i}{2}$

Simplify $\frac{12 \pm 2 i}{2}$ .

$x = 6 \pm i$

Change the $\pm$ to $-$ .

$x = 6 - i$

The final answer is the combination of both solutions.

$x = 6 + i, 6 - i$

Step 8

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

$(x - 3) (x - 6 - i) (x - 6 + i)$

Step 9

These are the roots (zeros) of the polynomial $x^{3} - 15 x^{2} + 73 x - 111$ .

$x = 3, 6 + i, 6 - i$

Step 10

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3-15x^2+73x-111

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