Precalculus Examples

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

Step 1

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

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

Step 2

Solve for $x$ .

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Step 2.1

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

$x - 2 = 0$

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

Step 2.2

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.2.1

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

$x - 2 = 0$

Step 2.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.3

Set ${(x - 1)}^{4}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

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

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

Step 2.3.2

Solve ${(x - 1)}^{4} = 0$ for $x$ .

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Step 2.3.2.1

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

$x - 1 = 0$

Step 2.3.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.4

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

$x = 2$ (Multiplicity of $1$ )

$x = 1$ (Multiplicity of $4$ )

$x = 2$ (Multiplicity of $1$ )

$x = 1$ (Multiplicity of $4$ )

Step 3