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Characteristics ofPolynomial FunctionsPrinta copy of this page.Many real-life situations can be modeled by polynomial functions.For example, the volume of a silo is a polynomial function of itsradius.In this lesson you will investigate the followingproperties ofpolynomial functions:A polynomial function is an equation of theform f(x)=anxn+an−1xn−1+...+a2x2+a1x+a0f(x)=anxn+an-1xn-1+...+a2x2+a1x+a0, where the coefficients anan, an−1an-1, ..., a1a1, a0a0 represent real numbers, anan is not zero, andthe exponents are non-negative integers.The real zeros of a function are the xx-intercepts of its graph.You can describe the behavior of a function based on itsdegree, the leading coefficient and the constant term.ReviewBefore beginning this lesson you should be able to:Use a graphing calculator to find thexandy-intercepts.
Analyze quadratic functions to identify characteristics of thecorresponding graph, including:overtexodomain and rangeodirection of openingoaxis of symmetryoxx- and yy-interceptsPolynomial Characteristics VideoNote: The video below covers the concepts from the lesson but isdifferent than the examples that follows.Examples and non-examples of polynomial functions include:ExamplesNon Examplesf(x)=5x3−8x2+x+1f(x)=5x3-8x2+x+1f(x)=xx2−3f(x)=xx2-3g(x)=6x3g(x)=6x3g(x)=2√ x−4x+5g(x)=2x-4x+5h(x)=2.1x4+3.5x3−4x2+x−7h(x)=2.1x4+3.5x3-4x2+x-7h(x)=√x−4h(x)=x-4Notice that a function CANNOT be classified as a polynomial functionif:an exponent is a fraction, negative number, radical, etc. Allexponents must be a positive integers.
a variable is in the denominator. All variables must exist in thenumerators only.a variable is located within a root signLet's look at a specific example to cover some important terms:Example 1f(x)=2x4+3x3−x2−5x+10f(x)=2x4+3x3-x2-5x+10.Thecoefficientsare 2, 3, -1, -5 and 10. They can be any realnumber.Theexponentsare 4, 3, 2 and 1. They must be positiveintegers.Thedegreeof the polynomial is 4. This is the highestexponent on a variable. The terms should always be arrangedsuch that the exponents are in descending order.Theleading coefficientis 2. It is the coefficient of thehighest degree term.

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Term
Fall
Professor
prof. whitener
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