Ch04+Polynomial+Functions

Chapter Four - Polynomial and Rational Functions

 * [[file:Chapter Four Syllabus.pdf]] lists this chapter's objectives, topics, homework assignments, and assessments.
 * [[file:Exponent Properties.pdf]] summarizes some important properties of exponential expressions.
 * [[file:Chapter Four Highlights.pdf]] shows the main points in Chapter Four in a set of not-so-pithy theorems.

4-1 Polynomial Functions

 * A polynomial in one variable, //x//, is an expression of the form //a 0 x n // + //a 1 x n-1 // + ... + //a n-2 x 2 // + //a n-1 x// + //a n //, where the coefficients //a 0 //, //a 1 //, //a 2 //, ...,//a n // are //complex numbers//, //a 0 // is not zero, and //n// represents a non-negative integer.
 * By that definition, a polynomial cannot have any roots (square or otherwise), nor can a polynomial have any negative powers of //x//.
 * The domain of any polynomial is the full number line.


 * Every polynomial equation with degree greater than zero has at least one root in the set of //complex numbers//.

4-2 Quadratic Equations and Inequalities

 * Standard form: //f//(//x//) = //ax// 2 + //bx// + //c//, where //a// does not equal zero. [By the way, what is the derivative, and where is it zero?]


 * Quadratic formula: //x =// [//–b// +/- sqrt//(b// 2 – 4//ac)// ] / (2//a//).
 * Now, starting with the standard form, see if you can derive this formula using the process of //completing the square//.


 * Quadratic inequalities can be solved by treating the quadratic polynomial curve as a functional boundary that divides the //x-y// plane into two regions. Test some point (//x//,//y//) on one side (or the other) of that boundary. If the test point satisfies the inequality, then //all// points on that side are solutions and that region should be shaded. Conversely, if the test point does //not// satisfy the inequality, then the solution lies on the //other// side of the curve. The boundary itself will not be part of the solution if the inequality is //**strict**// (i.e., > or <), so in graphing the solution the boundary should be shown as a **//dashed curve//**. If the inequality is **//non-strict//**, then draw the solution with a **//solid//** curve.

4-3 The Remainder and Factor Theorems

 * **Remainder Theorem**: By dividing a polynomial //P//(//x//) by (//x// – r), the remainder, //P//(//r//), is a constant and //P//(x) = (//x// – //r//) Q(//x//) + //P//(r).
 * //Q//(//x//) will be a polynomial of degree exactly one less than the degree of //P//(//x//).


 * **Factor Theorem**:
 * The binomial (//x// – //r//) is a factor of //P//(//x//) if and only if the remainder //P//(//r//) equals zero.

4-4 The Rational Root Theorem

 * Suppose you have a polynomial equation //P//(//x//) = //a// 0 //x// //n// + //a// //1// //x// //n-1// + ...+//a// //n-1// //x// +//a// //n// = 0, where all the coefficients are integers.The **Rational Root Theorem** basically says if this polynomial has any rational roots, then those roots must take the form //p/////q// where //p// is a factor of //a// //n//, and //q// is a factor of //a// 0.
 * There is an associated **Intergral Root Theorem** that is a special case of the Rational Root Theorem. Suppose //a 0 // = 1. Then from the RRT, //q// can take on values of +1 or –1 only. Consequently, if there are any rational roots, then they must be factors of //a// n.

Review for quiz on Sections 4.1 through 4.5:

Solution for review problems:

4-6 Rational Equations and Partial Fractions

 * [[file:Review 4-6 Solving Rational Equations.pdf]] is a set of practice problems with answers appended.

4-7 Radical Equations and Inequalities

 * [[file:Review 4-7 Radical Equations.pdf]] is a set of practice problems with answers provided.


 * Review for quiz on Sections 4.6 and 4.7: [[file:Review 4-6 to 4-7.pdf]]
 * Solutions to review problems: [[file:Review 4-6 to 4-7 solutions.pdf]]

Review for Chapter Four test:.

Solution for Chapter Four review:.