# Calculus Examples

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

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.

Simplify each term.

Raise to the power of .

Raise to the power of .

Multiply by .

Multiply by .

Subtract from .

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 AC method.

Factor using the AC method.

Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .

Write the factored form using these integers.

Remove unnecessary parentheses.

Set the factor equal to .

Add to both sides of the equation.

Set the factor equal to .

Add to both sides of the equation.

Set the factor equal to .

Subtract from both sides of the equation.

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 )

(Multiplicity of )

(Multiplicity of )