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

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

Step 1

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

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

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 $0$ .

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

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

Set ${(x + 8)}^{2}$ equal to $0$ and solve for $x$ .

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

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

Solve ${(x + 8)}^{2} = 0$ for $x$ .

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

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

Set ${(x - 8)}^{3}$ equal to $0$ and solve for $x$ .

Set ${(x - 8)}^{3}$ equal to $0$ .

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

Solve ${(x - 8)}^{3} = 0$ for $x$ .

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

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

The final solution is all the values that make $3 {(x + 8)}^{2} {(x - 8)}^{3} = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = - 8$ (Multiplicity of $2$ )

$x = 8$ (Multiplicity of $3$ )

$x = - 8$ (Multiplicity of $2$ )

$x = 8$ (Multiplicity of $3$ )

Step 3

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

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