Identify the Zeros and Their Multiplicities x^3-8x^2-x+8

$x^{3} - 8 x^{2} - x + 8$

Step 1

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

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

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.

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

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

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

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

$(x - 8) (x^{2} - 1) = 0$

Rewrite $1$ as $1^{2}$ .

$(x - 8) (x^{2} - 1^{2}) = 0$

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 $b = 1$ .

$(x - 8) ((x + 1) (x - 1)) = 0$

Remove unnecessary parentheses.

$(x - 8) (x + 1) (x - 1) = 0$

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

$x - 8 = 0$

$x + 1 = 0$

$x - 1 = 0$

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

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

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $(x - 8) (x + 1) (x - 1) = 0$ true. The multiplicity of a root is the number of times the root appears.

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

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

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

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

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

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

Step 3

Identify the Zeros and Their Multiplicities x^3-8x^2-x+8

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