## Key Concepts

- To find [latex]f\left(k\right)[/latex], determine the remainder of the polynomial [latex]f\left(x\right)[/latex] when it is divided by [latex]x-k[/latex].
*k*is a zero of [latex]f\left(x\right)[/latex] if and only if [latex]\left(x-k\right)[/latex] is a factor of [latex]f\left(x\right)[/latex].- Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient.
- When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
- Synthetic division can be used to find the zeros of a polynomial function.
- According to the Fundamental Theorem, every polynomial function has at least one complex zero.
- Every polynomial function with degree greater than 0 has at least one complex zero.
- Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form [latex]\left(x-c\right)[/latex], where
*c*is a complex number. - The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.
- The number of negative real zeros of a polynomial function is either the number of sign changes of [latex]f\left(-x\right)[/latex] or less than the number of sign changes by an even integer.
- Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division.

## Glossary

**Descartes’ Rule of Signs**- a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of [latex]f\left(x\right)[/latex] and [latex]f\left(-x\right)[/latex]

**Factor Theorem***k*is a zero of polynomial function [latex]f\left(x\right)[/latex] if and only if [latex]\left(x-k\right)[/latex] is a factor of [latex]f\left(x\right)[/latex]

**Fundamental Theorem of Algebra**- a polynomial function with degree greater than 0 has at least one complex zero

**Linear Factorization Theorem**- allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex]\left(x-c\right)[/latex], where
*c*is a complex number

**Rational Zero Theorem**- the possible rational zeros of a polynomial function have the form [latex]\frac{p}{q}[/latex] where
*p*is a factor of the constant term and*q*is a factor of the leading coefficient.

**Remainder Theorem**- if a polynomial [latex]f\left(x\right)[/latex] is divided by [latex]x-k[/latex], then the remainder is equal to the value [latex]f\left(k\right)[/latex]