Find the Roots/Zeros Using the Rational Roots Test f(x)=3x^3+7x^2-33x-77

$f (x) = 3 x^{3} + 7 x^{2} - 33 x - 77$

Step 1

Factor the left side of the equation.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{3} + 7 x^{2}) - 33 x - 77 = 0$

Factor out the greatest common factor (GCF) from each group.

$x^{2} (3 x + 7) - 11 (3 x + 7) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x + 7$ .

$(3 x + 7) (x^{2} - 11) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 x + 7 = 0$

$x^{2} - 11 = 0$

Step 3

Set $3 x + 7$ equal to $0$ and solve for $x$ .

Set $3 x + 7$ equal to $0$ .

$3 x + 7 = 0$

Solve $3 x + 7 = 0$ for $x$ .

Subtract $7$ from both sides of the equation.

$3 x = - 7$

Divide each term in $3 x = - 7$ by $3$ and simplify.

Divide each term in $3 x = - 7$ by $3$ .

$\frac{3 x}{3} = \frac{- 7}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 7}{3}$

Divide $x$ by $1$ .

$x = \frac{- 7}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{7}{3}$

Step 4

Set $x^{2} - 11$ equal to $0$ and solve for $x$ .

Set $x^{2} - 11$ equal to $0$ .

$x^{2} - 11 = 0$

Solve $x^{2} - 11 = 0$ for $x$ .

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 5

The final solution is all the values that make $(3 x + 7) (x^{2} - 11) = 0$ true.

$x = - \frac{7}{3}, \sqrt{11}, - \sqrt{11}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7}{3}, \sqrt{11}, - \sqrt{11}$

Decimal Form:

$x = - 2.\overline{3}, 3.31662479 \dots, - 3.31662479 \dots$

Mixed Number Form:

$x = - 2 \frac{1}{3}, \sqrt{11}, - \sqrt{11}$

Step 7

Find the Roots/Zeros Using the Rational Roots Test f(x)=3x^3+7x^2-33x-77

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