Find the Roots/Zeros Using the Rational Roots Test x^3-6x^2+61x-106

$x^{3} - 6 x^{2} + 61 x - 106$

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.

$p = \pm 1, \pm 2, \pm 53, \pm 106$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 53, \pm 106$

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.

${(2)}^{3} - 6 {(2)}^{2} + 61 (2) - 106$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 2$ is a root of the polynomial.

Simplify each term.

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

$8 - 6 {(2)}^{2} + 61 (2) - 106$

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

$8 - 6 \cdot 4 + 61 (2) - 106$

Multiply $- 6$ by $4$ .

$8 - 24 + 61 (2) - 106$

Multiply $61$ by $2$ .

$8 - 24 + 122 - 106$

Simplify by adding and subtracting.

Subtract $24$ from $8$ .

$- 16 + 122 - 106$

Add $- 16$ and $122$ .

$106 - 106$

Subtract $106$ from $106$ .

$0$

Step 5

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.

$\frac{x^{3} - 6 x^{2} + 61 x - 106}{x - 2}$

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.

$2$	$1$	$- 6$	$61$	$- 106$
	\xa0			\xa0

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

$2$	$1$	$- 6$	$61$	$- 106$
	\xa0			\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(2)$ and place the result of $(2)$ under the next term in the dividend $(- 6)$ .

$2$	$1$	$- 6$	$61$	$- 106$
	\xa0	$2$		\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.

$2$	$1$	$- 6$	$61$	$- 106$
	\xa0	$2$		\xa0
	$1$	$- 4$

Multiply the newest entry in the result $(- 4)$ by the divisor $(2)$ and place the result of $(- 8)$ under the next term in the dividend $(61)$ .

$2$	$1$	$- 6$	$61$	$- 106$
	\xa0	$2$	$- 8$	\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.

$2$	$1$	$- 6$	$61$	$- 106$
	\xa0	$2$	$- 8$	\xa0
	$1$	$- 4$	$53$

Multiply the newest entry in the result $(53)$ by the divisor $(2)$ and place the result of $(106)$ under the next term in the dividend $(- 106)$ .

$2$	$1$	$- 6$	$61$	$- 106$
	\xa0	$2$	$- 8$	$106$
	$1$	$- 4$	$53$

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

$2$	$1$	$- 6$	$61$	$- 106$
	\xa0	$2$	$- 8$	$106$
	$1$	$- 4$	$53$	$0$

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

$1 x^{2} + - 4 x + 53$

Simplify the quotient polynomial.

$x^{2} - 4 x + 53$

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 $a = 1$ , $b = - 4$ , and $c = 53$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 53)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 53}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 53$ .

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 53}}{2 \cdot 1}$

Multiply $- 4$ by $53$ .

$x = \frac{4 \pm \sqrt{16 - 212}}{2 \cdot 1}$

Subtract $212$ from $16$ .

$x = \frac{4 \pm \sqrt{- 196}}{2 \cdot 1}$

Rewrite $- 196$ as $- 1 (196)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 196}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (196)}$ as $\sqrt{- 1} \cdot \sqrt{196}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{196}}{2 \cdot 1}$

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

$x = \frac{4 \pm i \cdot \sqrt{196}}{2 \cdot 1}$

Rewrite $196$ as $14^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{14^{2}}}{2 \cdot 1}$

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

$x = \frac{4 \pm i \cdot 14}{2 \cdot 1}$

Move $14$ to the left of $i$ .

$x = \frac{4 \pm 14 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 14 i}{2}$

Simplify $\frac{4 \pm 14 i}{2}$ .

$x = 2 \pm 7 i$

The final answer is the combination of both solutions.

$x = 2 + 7 i, 2 - 7 i$

Step 8

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

$(x - 2) (x - 2 - 7 i) (x - 2 + 7 i)$

Step 9

These are the roots (zeros) of the polynomial $x^{3} - 6 x^{2} + 61 x - 106$ .

$x = 2, 2 + 7 i, 2 - 7 i$

Step 10

Find the Roots/Zeros Using the Rational Roots Test x^3-6x^2+61x-106

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