Algebra Examples

$f (x) = 6 {(x^{2} + 1)}^{2} {(x - 5)}^{3}$

Step 1

Set $6 {(x^{2} + 1)}^{2} {(x - 5)}^{3}$ equal to $0$ .

$6 {(x^{2} + 1)}^{2} {(x - 5)}^{3} = 0$

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 $0$ .

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

$x^{2} + 1 = 0$

Step 2.2.2.2

Solve for $x$ .

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Step 2.2.2.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 2.2.2.2.2

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

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

Step 2.2.2.2.3

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

$x = \pm i$

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.

$x = i$

Step 2.2.2.2.4.2

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

$x = - i$

Step 2.2.2.2.4.3

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

$x = i, - i$

Step 2.3

Set ${(x - 5)}^{3}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

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

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

Step 2.3.2

Solve ${(x - 5)}^{3} = 0$ for $x$ .

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Step 2.3.2.1

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

$x - 5 = 0$

Step 2.3.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.4

The final solution is all the values that make $6 {(x^{2} + 1)}^{2} {(x - 5)}^{3} = 0$ true. The multiplicity of a root is the number of times the root appears.

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

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

$x = 5$ (Multiplicity of $3$ )

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

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

$x = 5$ (Multiplicity of $3$ )

Step 3