Precalculus Examples

$2 x^{2} (x - 1) {(x + 2)}^{3} {(x^{2} + 1)}^{2}$

Step 1

Set $2 x^{2} (x - 1) {(x + 2)}^{3} {(x^{2} + 1)}^{2}$ equal to $0$ .

$2 x^{2} (x - 1) {(x + 2)}^{3} {(x^{2} + 1)}^{2} = 0$

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 $0$ .

$x^{2} = 0$

$x - 1 = 0$

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

${(x^{2} + 1)}^{2} = 0$

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

Solve $x^{2} = 0$ for $x$ .

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Step 2.2.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 2.2.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.2.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.2.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.2.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.3.1

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

$x - 1 = 0$

Step 2.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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Step 2.4.2.1

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

$x + 2 = 0$

Step 2.4.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.5

Set ${(x^{2} + 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.5.1

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

${(x^{2} + 1)}^{2} = 0$

Step 2.5.2

Solve ${(x^{2} + 1)}^{2} = 0$ for $x$ .

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Step 2.5.2.1

Set the $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Step 2.5.2.2

Solve for $x$ .

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Step 2.5.2.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 2.5.2.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1}$

Step 2.5.2.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 2.5.2.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.2.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 2.5.2.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 2.5.2.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 2.6

The final solution is all the values that make $2 x^{2} (x - 1) {(x + 2)}^{3} {(x^{2} + 1)}^{2} = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = 0$ (Multiplicity of $2$ )

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

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

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

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

$x = 0$ (Multiplicity of $2$ )

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

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

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

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

Step 3