Summary: Methods for Finding Zeros of Polynomial Functions

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]