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

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

Step 1

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

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

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 $0$ .

$x = 0$

$x - 5 = 0$

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

Set $x$ equal to $0$ .

$x = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set ${(x + 3)}^{2}$ equal to $0$ and solve for $x$ .

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

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

Solve ${(x + 3)}^{2} = 0$ for $x$ .

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

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

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

$x = 0$ (Multiplicity of $1$ )

$x = 5$ (Multiplicity of $1$ )

$x = - 3$ (Multiplicity of $2$ )

$x = 0$ (Multiplicity of $1$ )

$x = 5$ (Multiplicity of $1$ )

$x = - 3$ (Multiplicity of $2$ )

Step 3

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

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