Identify the Zeros and Their Multiplicities f(x)=x^3+8x^2-9x-72

$f (x) = x^{3} + 8 x^{2} - 9 x - 72$

Step 1

Set $x^{3} + 8 x^{2} - 9 x - 72$ equal to $0$ .

$x^{3} + 8 x^{2} - 9 x - 72 = 0$

Step 2

Solve for $x$ .

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.

$(x^{3} + 8 x^{2}) - 9 x - 72 = 0$

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

$x^{2} (x + 8) - 9 (x + 8) = 0$

Factor the polynomial by factoring out the greatest common factor, $x + 8$ .

$(x + 8) (x^{2} - 9) = 0$

Rewrite $9$ as $3^{2}$ .

$(x + 8) (x^{2} - 3^{2}) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$(x + 8) ((x + 3) (x - 3)) = 0$

Remove unnecessary parentheses.

$(x + 8) (x + 3) (x - 3) = 0$

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

$x + 8 = 0$

$x + 3 = 0$

$x - 3 = 0$

Set $x + 8$ equal to $0$ and solve for $x$ .

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

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

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

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

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.

$x = 3$

The final solution is all the values that make $(x + 8) (x + 3) (x - 3) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = - 8$ (Multiplicity of $1$ )

$x = - 3$ (Multiplicity of $1$ )

$x = 3$ (Multiplicity of $1$ )

$x = - 8$ (Multiplicity of $1$ )

$x = - 3$ (Multiplicity of $1$ )

$x = 3$ (Multiplicity of $1$ )

Step 3

Identify the Zeros and Their Multiplicities f(x)=x^3+8x^2-9x-72

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