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 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]
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