Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3+7x^2-11x-77

$f (x) = x^{3} + 7 x^{2} - 11 x - 77$

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.

$p = \pm 1, \pm 7, \pm 11, \pm 77$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 7, \pm 11, \pm 77$

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.

${(- 7)}^{3} + 7 {(- 7)}^{2} - 11 \cdot - 7 - 77$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 7$ is a root of the polynomial.

Simplify each term.

Raise $- 7$ to the power of $3$ .

$- 343 + 7 {(- 7)}^{2} - 11 \cdot - 7 - 77$

Raise $- 7$ to the power of $2$ .

$- 343 + 7 \cdot 49 - 11 \cdot - 7 - 77$

Multiply $7$ by $49$ .

$- 343 + 343 - 11 \cdot - 7 - 77$

Multiply $- 11$ by $- 7$ .

$- 343 + 343 + 77 - 77$

Simplify by adding and subtracting.

Add $- 343$ and $343$ .

$0 + 77 - 77$

Add $0$ and $77$ .

$77 - 77$

Subtract $77$ from $77$ .

$0$

Step 5

Since $- 7$ is a known root, divide the polynomial by $x + 7$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 7 x^{2} - 11 x - 77}{x + 7}$

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.

$- 7$	$1$	$7$	$- 11$	$- 77$
	\xa0			\xa0

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

$- 7$	$1$	$7$	$- 11$	$- 77$
	\xa0			\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 7)$ and place the result of $(- 7)$ under the next term in the dividend $(7)$ .

$- 7$	$1$	$7$	$- 11$	$- 77$
	\xa0	$- 7$		\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.

$- 7$	$1$	$7$	$- 11$	$- 77$
	\xa0	$- 7$		\xa0
	$1$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(- 7)$ and place the result of $(0)$ under the next term in the dividend $(- 11)$ .

$- 7$	$1$	$7$	$- 11$	$- 77$
	\xa0	$- 7$	$0$	\xa0
	$1$	$0$

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

$- 7$	$1$	$7$	$- 11$	$- 77$
	\xa0	$- 7$	$0$	\xa0
	$1$	$0$	$- 11$

Multiply the newest entry in the result $(- 11)$ by the divisor $(- 7)$ and place the result of $(77)$ under the next term in the dividend $(- 77)$ .

$- 7$	$1$	$7$	$- 11$	$- 77$
	\xa0	$- 7$	$0$	$77$
	$1$	$0$	$- 11$

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

$- 7$	$1$	$7$	$- 11$	$- 77$
	\xa0	$- 7$	$0$	$77$
	$1$	$0$	$- 11$	$0$

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

$1 x^{2} + 0 x - 11$

Simplify the quotient polynomial.

$x^{2} - 11$

Step 7

Solve the equation to find any remaining roots.

Add $11$ to both sides of the equation.

$x^{2} = 11$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{11}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{11}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{11}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{11}, - \sqrt{11}$

Step 8

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

$(x + 7) (x - \sqrt{11}) (x + \sqrt{11})$

Step 9

These are the roots (zeros) of the polynomial $x^{3} + 7 x^{2} - 11 x - 77$ .

$x = - 7, \sqrt{11}, - \sqrt{11}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - 7, \sqrt{11}, - \sqrt{11}$

Decimal Form:

$x = - 7, 3.31662479 \dots, - 3.31662479 \dots$

Step 11

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3+7x^2-11x-77

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