# Identify the Zeros and Their Multiplicities x^6-3x^5+4x^3

To find the roots/zeros, set equal to and solve.
Factor the left side of the equation.
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 .
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 and solve.
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 and solve.
Set the next factor equal to .
Subtract from both sides of the equation.
Set the next factor equal to and solve.
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