Precalculus Examples

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

Step 1

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

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

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 $0$ .

$x = 0$

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

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

Step 2.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

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Step 2.4.1

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

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

Step 2.4.2

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

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Step 2.4.2.1

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

$x + 5 = 0$

Step 2.4.2.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 2.5

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

$x = 0$ (Multiplicity of $1$ )

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

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

$x = 0$ (Multiplicity of $1$ )

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

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

Step 3