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 of Algebra, 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]