Identify the Zeros and Their Multiplicities 5(x+7)^2(x-7)^3

$5 {(x + 7)}^{2} {(x - 7)}^{3}$

Step 1

Set $5 {(x + 7)}^{2} {(x - 7)}^{3}$ equal to $0$ .

$5 {(x + 7)}^{2} {(x - 7)}^{3} = 0$

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 $0$ .

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

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

Set ${(x + 7)}^{2}$ equal to $0$ and solve for $x$ .

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

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

Solve ${(x + 7)}^{2} = 0$ for $x$ .

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

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

Set ${(x - 7)}^{3}$ equal to $0$ and solve for $x$ .

Set ${(x - 7)}^{3}$ equal to $0$ .

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

Solve ${(x - 7)}^{3} = 0$ for $x$ .

Set the $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

The final solution is all the values that make $5 {(x + 7)}^{2} {(x - 7)}^{3} = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = - 7$ (Multiplicity of $2$ )

$x = 7$ (Multiplicity of $3$ )

$x = - 7$ (Multiplicity of $2$ )

$x = 7$ (Multiplicity of $3$ )

Step 3

Identify the Zeros and Their Multiplicities 5(x+7)^2(x-7)^3

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