Finite Math Examples

$f (x) = - 4 x^{2} (x - 9) {(x + 2)}^{3}$

Step 1

Set $- 4 x^{2} (x - 9) {(x + 2)}^{3}$ equal to $0$ .

$- 4 x^{2} (x - 9) {(x + 2)}^{3} = 0$

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 $0$ .

$x^{2} = 0$

$x - 9 = 0$

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

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 $x - 9$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 2.3.2

Add $9$ to both sides of the equation.

$x = 9$

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

The final solution is all the values that make $- 4 x^{2} (x - 9) {(x + 2)}^{3} = 0$ true. The multiplicity of a root is the number of times the root appears.

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

$x = 9$ (Multiplicity of $1$ )

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

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

$x = 9$ (Multiplicity of $1$ )

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

Step 3