# Calculus Examples

Identify the Zeros and Their Multiplicities
To find the roots/zeros of the function, set the function equal to and solve.
Factor the left side of the equation.
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 .
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 equal to and solve for .
Set the factor equal to .
Add to both sides of the equation.
Set equal to and solve for .
Set the factor equal to .
Add to both sides of the equation.
Set equal to and solve for .
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 )