Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3+3x^2-63x-65

$f (x) = x^{3} + 3 x^{2} - 63 x - 65$

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.

$p = \pm 1, \pm 5, \pm 13, \pm 65$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 5, \pm 13, \pm 65$

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.

${(- 1)}^{3} + 3 {(- 1)}^{2} - 63 \cdot - 1 - 65$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 1$ is a root of the polynomial.

Simplify each term.

Raise $- 1$ to the power of $3$ .

$- 1 + 3 {(- 1)}^{2} - 63 \cdot - 1 - 65$

Raise $- 1$ to the power of $2$ .

$- 1 + 3 \cdot 1 - 63 \cdot - 1 - 65$

Multiply $3$ by $1$ .

$- 1 + 3 - 63 \cdot - 1 - 65$

Multiply $- 63$ by $- 1$ .

$- 1 + 3 + 63 - 65$

Simplify by adding and subtracting.

Add $- 1$ and $3$ .

$2 + 63 - 65$

Add $2$ and $63$ .

$65 - 65$

Subtract $65$ from $65$ .

$0$

Step 5

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

$\frac{x^{3} + 3 x^{2} - 63 x - 65}{x + 1}$

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.

$- 1$	$1$	$3$	$- 63$	$- 65$
	\xa0			\xa0

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

$- 1$	$1$	$3$	$- 63$	$- 65$
	\xa0			\xa0
	$1$

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

$- 1$	$1$	$3$	$- 63$	$- 65$
	\xa0	$- 1$		\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.

$- 1$	$1$	$3$	$- 63$	$- 65$
	\xa0	$- 1$		\xa0
	$1$	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(- 1)$ and place the result of $(- 2)$ under the next term in the dividend $(- 63)$ .

$- 1$	$1$	$3$	$- 63$	$- 65$
	\xa0	$- 1$	$- 2$	\xa0
	$1$	$2$

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

$- 1$	$1$	$3$	$- 63$	$- 65$
	\xa0	$- 1$	$- 2$	\xa0
	$1$	$2$	$- 65$

Multiply the newest entry in the result $(- 65)$ by the divisor $(- 1)$ and place the result of $(65)$ under the next term in the dividend $(- 65)$ .

$- 1$	$1$	$3$	$- 63$	$- 65$
	\xa0	$- 1$	$- 2$	$65$
	$1$	$2$	$- 65$

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

$- 1$	$1$	$3$	$- 63$	$- 65$
	\xa0	$- 1$	$- 2$	$65$
	$1$	$2$	$- 65$	$0$

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

$1 x^{2} + 2 x - 65$

Simplify the quotient polynomial.

$x^{2} + 2 x - 65$

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 $a = 1$ , $b = 2$ , and $c = - 65$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot - 65)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 65}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 65$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 65}}{2 \cdot 1}$

Multiply $- 4$ by $- 65$ .

$x = \frac{- 2 \pm \sqrt{4 + 260}}{2 \cdot 1}$

Add $4$ and $260$ .

$x = \frac{- 2 \pm \sqrt{264}}{2 \cdot 1}$

Rewrite $264$ as $2^{2} \cdot 66$ .

Factor $4$ out of $264$ .

$x = \frac{- 2 \pm \sqrt{4 (66)}}{2 \cdot 1}$

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 66}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{66}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{66}}{2}$

Simplify $\frac{- 2 \pm 2 \sqrt{66}}{2}$ .

$x = - 1 \pm \sqrt{66}$

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

Simplify the numerator.

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 65}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 65$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 65}}{2 \cdot 1}$

Multiply $- 4$ by $- 65$ .

$x = \frac{- 2 \pm \sqrt{4 + 260}}{2 \cdot 1}$

Add $4$ and $260$ .

$x = \frac{- 2 \pm \sqrt{264}}{2 \cdot 1}$

Rewrite $264$ as $2^{2} \cdot 66$ .

Factor $4$ out of $264$ .

$x = \frac{- 2 \pm \sqrt{4 (66)}}{2 \cdot 1}$

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 66}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{66}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{66}}{2}$

Simplify $\frac{- 2 \pm 2 \sqrt{66}}{2}$ .

$x = - 1 \pm \sqrt{66}$

Change the $\pm$ to $+$ .

$x = - 1 + \sqrt{66}$

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

Simplify the numerator.

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 65}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 65$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 65}}{2 \cdot 1}$

Multiply $- 4$ by $- 65$ .

$x = \frac{- 2 \pm \sqrt{4 + 260}}{2 \cdot 1}$

Add $4$ and $260$ .

$x = \frac{- 2 \pm \sqrt{264}}{2 \cdot 1}$

Rewrite $264$ as $2^{2} \cdot 66$ .

Factor $4$ out of $264$ .

$x = \frac{- 2 \pm \sqrt{4 (66)}}{2 \cdot 1}$

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 66}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{66}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{66}}{2}$

Simplify $\frac{- 2 \pm 2 \sqrt{66}}{2}$ .

$x = - 1 \pm \sqrt{66}$

Change the $\pm$ to $-$ .

$x = - 1 - \sqrt{66}$

The final answer is the combination of both solutions.

$x = - 1 + \sqrt{66}, - 1 - \sqrt{66}$

Step 8

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

$(x + 1) (x + 1 - \sqrt{66}) (x + 1 + \sqrt{66})$

Step 9

These are the roots (zeros) of the polynomial $x^{3} + 3 x^{2} - 63 x - 65$ .

$x = - 1, - 1 + \sqrt{66}, - 1 - \sqrt{66}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - 1, - 1 + \sqrt{66}, - 1 - \sqrt{66}$

Decimal Form:

$x = - 1, 7.12403840 \dots, - 9.12403840 \dots$

Step 11

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3+3x^2-63x-65

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