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 an Algebra 2 worksheet in the page, take this from Introduction to Polynomial Functions through Complex Zeros and Fundamental Theorem of Algebra to Modeling with Polynomial Functions. Every practice problem is scored as it is answered, so a zero counted once when its multiplicity is two shows up immediately. 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 build up to reading zeros, multiplicity, and turning points off a curve. Complex Zeros and Fundamental Theorem of Algebra accounts for every root a polynomial has, Rational Zeros Theorem and Polynomial Equations gives a way to find the rational ones, and Irrational Conjugate Theorem and Descartes' Rule narrows the search further. Modeling with Polynomial Functions applies all of it to data. It sits in Algebra (A) and Functions (F) in the high school Common Core, with complex zeros reaching into Number and Quantity (N).
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.