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

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

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.

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 .

Take the cube root of both sides of the equation to eliminate the exponent on the left side.

Simplify .

Rewrite as .

Pull terms out from under the radical, assuming real numbers.

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 x^6-3x^5+4x^3