Find the Roots (Zeros) f(x)=8x^7(x^2+x-12)

$f (x) = 8 x^{7} (x^{2} + x - 12)$

Step 1

Set $8 x^{7} (x^{2} + x - 12)$ equal to $0$ .

$8 x^{7} (x^{2} + x - 12) = 0$

Step 2

Solve for $x$ .

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

$x^{7} = 0$

$x^{2} + x - 12 = 0$

Set $x^{7}$ equal to $0$ and solve for $x$ .

Set $x^{7}$ equal to $0$ .

$x^{7} = 0$

Solve $x^{7} = 0$ for $x$ .

Take the 7th root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[7]{0}$

Simplify $\sqrt[7]{0}$ .

Rewrite $0$ as $0^{7}$ .

$x = \sqrt[7]{0^{7}}$

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

$x = 0$

Set $x^{2} + x - 12$ equal to $0$ and solve for $x$ .

Set $x^{2} + x - 12$ equal to $0$ .

$x^{2} + x - 12 = 0$

Solve $x^{2} + x - 12 = 0$ for $x$ .

Factor $x^{2} + x - 12$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $1$ .

$- 3, 4$

Write the factored form using these integers.

$(x - 3) (x + 4) = 0$

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

$x - 3 = 0$

$x + 4 = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $(x - 3) (x + 4) = 0$ true.

$x = 3, - 4$

The final solution is all the values that make $8 x^{7} (x^{2} + x - 12) = 0$ true.

$x = 0, 3, - 4$

Step 3

Find the Roots (Zeros) f(x)=8x^7(x^2+x-12)

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