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

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 .

Multiply by .

Raise to the power of .

Multiply by .

Subtract from .

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.

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 .

Factor out of .

Factor out of .

Factor out of .

Rewrite as .

Factor out of .

Factor out of .

Multiply each term in by

Multiply each term in by .

Simplify .

Apply the distributive property.

Simplify.

Multiply by .

Multiply by .

Apply the distributive property.

Simplify.

Multiply .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Use the quadratic formula to find the solutions.

Substitute the values , , and into the quadratic formula and solve for .

Simplify.

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Subtract from .

Rewrite as .

Rewrite as .

Rewrite as .

Rewrite as .

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

Move to the left of .

Multiply by .

Simplify .

Simplify the expression to solve for the portion of the .

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Subtract from .

Rewrite as .

Rewrite as .

Rewrite as .

Rewrite as .

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

Move to the left of .

Multiply by .

Simplify .

Change the to .

Simplify the expression to solve for the portion of the .

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Subtract from .

Rewrite as .

Rewrite as .

Rewrite as .

Rewrite as .

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

Move to the left of .

Multiply by .

Simplify .

Change the to .

The final answer is the combination of both solutions.

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 f(x)=-2x^3-11x^2-110x+58