Chapter 3: Polynomial and Rational Functions
Chapter 3 goes beyond linear and quadratic functions to polynomials of any degree and rational functions (ratios of polynomials). You'll learn to find zeros, graph behavior, and understand asymptotes. These skills connect algebra to calculus.
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Sections
3.1Quadratic Functions and Models
A quadratic function has the form f(x) = ax² + bx + c. Its graph is a parabola — symmetric, with a single turning point called the vertex. The vertex tells you the maximum or minimum value of the function.
3.2Polynomial Functions and Their Graphs
This section builds intuition for the shape and behavior of polynomial graphs. The degree and leading coefficient determine the end behavior; the zeros determine where the graph crosses the x-axis. Between knowing end behavior and zeros, you can sketch any polynomial.
3.3Dividing Polynomials
Long division and synthetic division let you divide polynomials to find zeros and factors. The Remainder Theorem and Factor Theorem make this powerful: they connect polynomial evaluation and factoring in one clean idea.
3.4Real Zeros of Polynomials
Finding all zeros of a polynomial can be tricky. This section gives you three tools: the Rational Zero Theorem (a list of candidates), Descartes' Rule of Signs (how many positive/negative zeros to expect), and the Upper/Lower Bound tests (to eliminate candidates faster).
3.5Complex Zeros and the Fundamental Theorem
The Fundamental Theorem of Algebra guarantees that every polynomial of degree n has exactly n zeros (counting multiplicity), as long as we allow complex numbers. Complex zeros always come in conjugate pairs for polynomials with real coefficients.
3.6Rational Functions
A rational function is the ratio of two polynomials: r(x) = P(x)/Q(x). These functions have asymptotes — vertical, horizontal, or oblique — that define the 'skeleton' of the graph. Understanding asymptotes is essential for calculus.
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