Algebra Examples

$f (x) = 5 {(x^{2} + 4)}^{2} {(x - 3)}^{3}$

Step 1

Set $5 {(x^{2} + 4)}^{2} {(x - 3)}^{3}$ equal to $0$ .

$5 {(x^{2} + 4)}^{2} {(x - 3)}^{3} = 0$

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 $0$ .

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

Set the $x^{2} + 4$ equal to $0$ .

$x^{2} + 4 = 0$

Step 2.2.2.2

Solve for $x$ .

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Step 2.2.2.2.1

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 2.2.2.2.2

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

$x = \pm \sqrt{- 4}$

Step 2.2.2.2.3

Simplify $\pm \sqrt{- 4}$ .

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Step 2.2.2.2.3.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 2.2.2.2.3.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 2.2.2.2.3.3

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

$x = \pm i \cdot \sqrt{4}$

Step 2.2.2.2.3.4

Rewrite $4$ as $2^{2}$ .

$x = \pm i \cdot \sqrt{2^{2}}$

Step 2.2.2.2.3.5

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

$x = \pm i \cdot 2$

Step 2.2.2.2.3.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 2.2.2.2.4

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

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Step 2.2.2.2.4.1

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

$x = 2 i$

Step 2.2.2.2.4.2

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

$x = - 2 i$

Step 2.2.2.2.4.3

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

$x = 2 i, - 2 i$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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Step 2.3.2.1

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

$x - 3 = 0$

Step 2.3.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.4

The final solution is all the values that make $5 {(x^{2} + 4)}^{2} {(x - 3)}^{3} = 0$ true. The multiplicity of a root is the number of times the root appears.

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

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

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

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

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

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

Step 3