Precalculus Examples

Identify the Zeros and Their Multiplicities
Step 1
Set equal to .
Step 2
Solve for .
Step 2.1
Factor the left side of the equation.
Step 2.1.1
Regroup terms.
Step 2.1.2
Rewrite as .
Step 2.1.3
Since both terms are perfect cubes, factor using the sum of cubes formula, where and .
Step 2.1.4
Simplify.
Step 2.1.4.1
Multiply by .
Step 2.1.4.2
Raise to the power of .
Step 2.1.5
Factor out of .
Step 2.1.5.1
Factor out of .
Step 2.1.5.2
Factor out of .
Step 2.1.5.3
Factor out of .
Step 2.1.6
Factor out of .
Step 2.1.6.1
Factor out of .
Step 2.1.6.2
Factor out of .
Step 2.1.7
Step 2.1.8
Factor using the perfect square rule.
Step 2.1.8.1
Rewrite as .
Step 2.1.8.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 2.1.8.3
Rewrite the polynomial.
Step 2.1.8.4
Factor using the perfect square trinomial rule , where and .
Step 2.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.3
Set equal to and solve for .
Step 2.3.1
Set equal to .
Step 2.3.2
Subtract from both sides of the equation.
Step 2.4
The final solution is all the values that make true. The multiplicity of a root is the number of times the root appears.
(Multiplicity of )
(Multiplicity of )
Step 3