Polynomial Functions
Lessons in this chapter
About Polynomial Functions
Polynomial Functions builds a complete picture of a polynomial’s graph and its zeros, real, irrational, and complex alike.
Introduction to Polynomial Functions defines a polynomial function and its basic vocabulary, before Key Features of Graphs 1 and Key Features of Graphs 2 cover reading and sketching a polynomial’s graph from its degree, end behavior, and zeros.
Complex Zeros and Fundamental Theorem of Algebra guarantees every polynomial has as many zeros (real or complex) as its degree, while Rational Zeros Theorem and Polynomial Equations gives a systematic way to test for possible rational zeros.
Irrational Conjugate Theorem and Descartes’ Rule add two more tools for narrowing down a polynomial’s zeros, and Modeling with Polynomial Functions closes the chapter by fitting a polynomial function to real data or a real situation.
Seven lessons, each a practice worksheet in the page, run from Introduction to Polynomial Functions through Rational Zeros Theorem and Polynomial Equations to Modeling with Polynomial Functions. The problems are scored as you answer them, so a zero missed in the list of candidates shows up at once. Free to try, with no login needed for your first 3.
Frequently asked questions
Introduction to Polynomial Functions sets up degree, leading coefficient, and end behavior, and Key Features of Graphs 1 and 2 cover zeros, multiplicity, and turning points. Complex Zeros and Fundamental Theorem of Algebra accounts for every root a polynomial has, Rational Zeros Theorem and Polynomial Equations narrows down which rational roots are worth testing, and Irrational Conjugate Theorem and Descartes' Rule narrows it further. Modeling with Polynomial Functions applies the whole set to data. This sits in Algebra (A) and Functions (F) in the high school Common Core, and in Unit 1 of the AP Precalculus course framework, Polynomial and Rational Functions.
A polynomial function is built from terms with non-negative integer exponents, and its degree (the highest exponent) determines its overall shape and end behavior.
End behavior describes what a graph does as x goes to positive or negative infinity, and it's determined entirely by the polynomial's degree and leading coefficient.
The Fundamental Theorem of Algebra guarantees a polynomial of degree n has exactly n zeros, counting complex zeros and repeated zeros.
The Rational Zeros Theorem lists every possible rational zero a polynomial could have, based on its leading coefficient and constant term, narrowing down what to test.
It says complex zeros and irrational zeros (of the form a + b√c) always come in conjugate pairs for a polynomial with real coefficients: find one, and you automatically know its pair.