Chapter 5: Polynomial & Rational Functions

Section 5.1 – Polynomial Functions & Models

Polynomial functions are a central topic in algebra, describing equations involving powers of variables with real coefficients. Understanding their structure and behavior is essential for modeling and solving real-world problems.

• Definition: A polynomial function is of the form , where and is a non-negative integer.

• Degree: The highest power of in the polynomial; determines the end behavior.

• Leading Coefficient: The coefficient of the highest degree term; affects the direction of the graph as .

Example 1: Identify end behavior of .

• Degree: 4 (even)

• Leading coefficient: 3 (positive)

• End Behavior: As , (both ends rise).

Section 5.2 – Properties of Rational Functions

Rational functions are quotients of polynomials and exhibit unique properties such as asymptotes and domain restrictions.

• Definition: , where .

• Domain: All real numbers except where .

• Vertical Asymptotes: Occur at zeros of (where denominator is zero and not canceled by numerator).

• Horizontal Asymptotes: Determined by degrees of numerator () and denominator ():

  • If , horizontal asymptote at .

  • If , horizontal asymptote at (ratio of leading coefficients).

  • If , no horizontal asymptote (may have an oblique/slant asymptote).

Example 2:

• Vertical asymptotes at , (denominator zeros).

• Horizontal asymptote at (degree numerator < degree denominator).

Section 5.3 – Graphing Rational Functions

Graphing rational functions involves analyzing their algebraic structure to determine intercepts, asymptotes, and overall shape.

1. Simplify the function and identify holes (common factors in numerator and denominator).

2. Find intercepts (set numerator and denominator to zero).

3. Identify vertical and horizontal (or slant) asymptotes.

4. Test intervals to determine the function's sign and shape between asymptotes.

Example 3:

• Hole at (factor cancels).

• Vertical asymptote at .

• Horizontal asymptote at (degrees equal, leading coefficients both 1).

Section 5.4 – Polynomial & Rational Inequalities

Solving inequalities involving polynomials and rational functions requires finding critical points and testing intervals.

1. Factor numerator and denominator.

2. Find critical points (where numerator or denominator is zero).

3. Test intervals between critical points to determine where the inequality holds.

Example 4: Solve

• Critical points:

• Test intervals:

• Solution: ,

Section 5.5 – Real Zeros of Polynomial Functions

Finding real zeros involves the Rational Root Theorem and synthetic division to test possible rational roots.

• Rational Root Theorem: Possible rational zeros of are , where divides and divides .

• Use synthetic division to test possible roots.

Example 5: Find a real root of

• Possible rational roots:

• Test using synthetic division to find actual zeros.

Section 5.6 – Complex Zeros & Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant polynomial has at least one complex root. Complex zeros occur in conjugate pairs if the polynomial has real coefficients.

• Fundamental Theorem of Algebra (FTA): Every polynomial of degree has exactly complex roots (counting multiplicities).

• Complex zeros of real polynomials occur in conjugate pairs.

Example 6: Solve

• Solution: