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Chapter 7

Polynomial Functions and Equations

A polynomial function is one in which the coefficients are all real numbers and the exponents on the variables are all whole numbers. A polynomial whose greatest power is 2 is called a quadratic polynomial; if the highest power is 3, then it’s called a cubic polynomial. A highest power of 4 earns the name quartic (not to be confused with quadratic), and a highest power of 5 is called quintic. There are more names for higher powers, but the usual practice is just to refer to the power rather than to try to come up with the Latin or Greek prefix.

The Problems You’ll Work On

In this chapter, you’ll work with polynomial functions and equations in the following ways:

 Determining the x and y intercepts from the function rule (equation)

 Solving polynomial equations using grouping

 Applying the rational root theorem to find roots

 Using Descartes’ rule of sign to count possible real roots

 Making use of synthetic division

 Graphing polynomial functions

What to Watch Out For

Don’t let common mistakes trip you up; watch for the following ones when working with polynomial functions and equations:

 Forgetting to change the signs in the factored form when identifying x-intercepts

 Making errors when simplifying the terms in f(–x) applying Descartes’ rule of sign

 Not changing the sign of the divisor when using synthetic division

 Not distinguishing between curves that cross from those that just touch the x-axis at an intercept

 Graphing the incorrect end-behavior on the right and left of the graphs

Recognizing the Intercepts of Polynomials

341–350 Find the intercepts of the polynomial.

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Factoring by Grouping to Solve for Intercepts

351–360 Find the intercepts of the polynomial. To find the x-intercepts, use factoring by grouping.

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Applying the Rational Root Theorem to Find Roots

361–370 Use the Rational Root Theorem and Descartes’ Rule of Signs to list the possible rational roots of the polynomial.

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Performing Synthetic Division to Factor Polynomials

371–380 Factor the polynomial expressions using synthetic division.

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Evaluating Polynomials for Input Values

381–390 Evaluate the functions for the given input using the remainder theorem.

381. Given , find f(1).

382. Given , find f(1).

383. Given , find f(1).

384. Given , find f(–1).

385. Given , find f(2).

386. Given , find f(–2).

387. Given , find f(–1).

388. Given , find f(–2).

389. Given , find f(1).

390. Given , find f(2).

Investigating End-Behavior of Polynomials

391–400 Determine the end-behavior of the polynomials.

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Sketching the Graphs of Polynomial Functions

401 – 410 Sketch the graph of the polynomial.

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