Precalculus Examples

Identify the Zeros and Their Multiplicities
To find the roots/zeros of the function, set the function equal to and solve.
Since does not contain the variable to solve for, move it to the right side of the equation by subtracting from both sides.
Move to the left side of the equation by adding it to both sides.
Factor the left side of the equation.
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Rewrite as .
Since both terms are perfect cubes, factor using the sum of cubes formula, where and .
Simplify.
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Multiply by to get .
Raise to the power of to get .
Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Factor out of .
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Reorder and .
Factor out of .
Add and to get .
Factor using the perfect square rule.
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Rewrite as .
Check the middle term by multiplying and compare this result with the middle term in the original expression.
Simplify.
Factor using the perfect square trinomial rule , where and .
Set equal to and solve for .
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Set the factor equal to .
Since does not contain the variable to solve for, move it to the right side of the equation by subtracting from both sides.
Solve the equations for . The multiplicity of a root is the number of times the root appears. For example, a factor of would have a root at with multiplicity of .
(Multiplicity of )
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