Key Concepts & Glossary
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]
Licenses & Attributions
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- Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..