# Polynomial Operations and Theorems

## Overview

### Description

Polynomial operations include addition, subtraction, multiplication, and division. Factoring polynomials may be useful when solving polynomial equations. Theorems about polynomial functions are used to find information about a function that is helpful when sketching a graph of the function.

### At A Glance

• Polynomials can be added, subtracted, multiplied, factored, and divided.
• Polynomials are added or subtracted by combining like terms.
• Polynomials are multiplied by using the distributive property and properties of exponents.
• Polynomials are factored by using a variety of techniques, including grouping and formulas.
• Polynomials are divided by factoring or using algorithms for long division or synthetic division.
• The factored form of a polynomial expression can be used to solve the related equation.
• If the polynomial that defines a function $f(x)$ is divided by $x - c$, the remainder is $f(c)$.
• According to the factor theorem, for a polynomial that defines a function $f(x)$, $x - c$ is a factor if and only if $f(c) = 0$.
• If $\frac{p}{q}$ is a rational zero of a polynomial function $f(x)$, then $q$ is a factor of the leading coefficient of the polynomial, and $p$ is a factor of the constant term of the polynomial.
• For a polynomial function $f(x)$, if $a < b$ and $f(a)$ and $f(b)$ have opposite signs, there is at least one real zero of $f(x)$ between $a$ and $b$.
• Every polynomial function $f(x)$ of degree $n>0$ has at least one complex zero.