Identify the Zeros and Their Multiplicities f(x)=x(x-1)(x-2)^4

$f (x) = x (x - 1) {(x - 2)}^{4}$

Step 1

Set $x (x - 1) {(x - 2)}^{4}$ equal to $0$ .

$x (x - 1) {(x - 2)}^{4} = 0$

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 $0$ .

$x = 0$

$x - 1 = 0$

${(x - 2)}^{4} = 0$

Set $x$ equal to $0$ .

$x = 0$

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.

$x = 1$

Set ${(x - 2)}^{4}$ equal to $0$ and solve for $x$ .

Set ${(x - 2)}^{4}$ equal to $0$ .

${(x - 2)}^{4} = 0$

Solve ${(x - 2)}^{4} = 0$ for $x$ .

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $x (x - 1) {(x - 2)}^{4} = 0$ true. The multiplicity of a root is the number of times the root appears.

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

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

$x = 2$ (Multiplicity of $4$ )

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

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

$x = 2$ (Multiplicity of $4$ )

Step 3

Identify the Zeros and Their Multiplicities f(x)=x(x-1)(x-2)^4

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