To find the roots/zeros, set equal to and solve.

Regroup terms.

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor.

Factor by grouping.

For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .

Factor out of .

Rewrite as plus

Apply the distributive property.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

Factor out the greatest common factor (GCF) from each group.

Factor the polynomial by factoring out the greatest common factor, .

Remove unnecessary parentheses.

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Apply the distributive property.

Multiply by by adding the exponents.

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Move to the left of .

Reorder terms.

Factor.

Rewrite in a factored form.

Factor using the rational roots test.

If a polynomial function has integer coefficients, then every rational zero will have the form where is a factor of the constant and is a factor of the leading coefficient.

Find every combination of . These are the possible roots of the polynomial function.

Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.

Substitute into the polynomial.

Raise to the power of .

Raise to the power of .

Multiply by .

Subtract from .

Add and .

Since is a known root, divide the polynomial by to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

Divide by .

Write as a set of factors.

Factor using the perfect square rule.

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 .

Remove unnecessary parentheses.

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .

Set the first factor equal to .

Add to both sides of the equation.

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Set the next factor equal to .

Subtract from both sides of the equation.

Set the next factor equal to .

Set the equal to .

Add to both sides of the equation.

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 )

(Multiplicity of )

Identify the Zeros and Their Multiplicities 4x^4-15x^3+9x^2+16x-12