Algebra Examples

$f (x) = {(x + 7)}^{2} (2 x + 1) {(x - 4)}^{3}$

Step 1

Set ${(x + 7)}^{2} (2 x + 1) {(x - 4)}^{3}$ equal to $0$ .

${(x + 7)}^{2} (2 x + 1) {(x - 4)}^{3} = 0$

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 $0$ .

${(x + 7)}^{2} = 0$

$2 x + 1 = 0$

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

Step 2.2

Set ${(x + 7)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.2.1

Set ${(x + 7)}^{2}$ equal to $0$ .

${(x + 7)}^{2} = 0$

Step 2.2.2

Solve ${(x + 7)}^{2} = 0$ for $x$ .

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Step 2.2.2.1

Set the $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 2.2.2.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 2.3

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 2.3.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 2.3.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 2.3.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 2.3.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 2.3.2.2.3

Simplify the right side.

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Step 2.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 2.4

Set ${(x - 4)}^{3}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

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

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

Step 2.4.2

Solve ${(x - 4)}^{3} = 0$ for $x$ .

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Step 2.4.2.1

Set the $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.4.2.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.5

The final solution is all the values that make ${(x + 7)}^{2} (2 x + 1) {(x - 4)}^{3} = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = - 7$ (Multiplicity of $2$ )

$x = - \frac{1}{2}$ (Multiplicity of $1$ )

$x = 4$ (Multiplicity of $3$ )

$x = - 7$ (Multiplicity of $2$ )

$x = - \frac{1}{2}$ (Multiplicity of $1$ )

$x = 4$ (Multiplicity of $3$ )

Step 3