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)f(x) is divided by xcx - c, the remainder is f(c)f(c).
  • According to the factor theorem, for a polynomial that defines a function f(x)f(x), xcx - c is a factor if and only if f(c)=0f(c) = 0.
  • If pq\frac{p}{q} is a rational zero of a polynomial function f(x)f(x), then qq is a factor of the leading coefficient of the polynomial, and pp is a factor of the constant term of the polynomial.
  • For a polynomial function f(x)f(x), if a<ba < b and f(a)f(a) and f(b)f(b) have opposite signs, there is at least one real zero of f(x)f(x) between aa and bb.
  • Every polynomial function f(x)f(x) of degree n>0n>0 has at least one complex zero.