BackChapter 5 Study Guide: Polynomial & Rational Functions
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Polynomial & Rational Functions
Section 5.1 – Polynomial Functions & Models
Polynomial functions are a central topic in algebra, describing equations involving powers of x with real coefficients. Understanding their structure and behavior is essential for modeling real-world phenomena and solving equations.
Definition: A polynomial function is of the form , where and is a non-negative integer.
Degree: The degree of a polynomial is the highest power of x present.
Leading Coefficient: The coefficient of the term with the highest degree.
End Behavior: Determined by the degree and leading coefficient.
Example 1: Identify end behavior of .
Degree: 4 (even)
Leading coefficient: 3 (positive)
End behavior: As , .
Section 5.2 – Properties of Rational Functions
Rational functions are quotients of polynomials and exhibit unique properties such as asymptotes and domain restrictions. Analyzing these properties is crucial for graphing and solving equations.
Definition: A rational function is , where .
Domain: All real numbers except where .
Vertical Asymptotes: Occur at zeros of (where denominator is zero).
Horizontal Asymptotes: Determined by degrees of numerator and denominator:
Degree of P(x) | Degree of Q(x) | Horizontal Asymptote |
|---|---|---|
n < m | m | |
n = m | m | (ratio of leading coefficients) |
n > m | m | None (may have oblique/slant asymptote) |
Example 2: has as horizontal asymptote.
Section 5.3 – Graphing Rational Functions
Graphing rational functions involves identifying key features such as intercepts, asymptotes, and the overall shape. A systematic approach helps in sketching accurate graphs.
1. Simplify function; identify holes (common factors in numerator and denominator).
2. Find intercepts (set for y-intercept, set numerator to zero for x-intercepts).
3. Identify asymptotes (vertical and horizontal).
4. Test intervals to determine shape.
Example 3:
Hole at (common 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 to determine where the inequality holds.
1. Factor numerator and denominator.
2. Find critical points (zeros of numerator and denominator).
3. Test intervals between critical points.
Example 4: Solve .
Critical points: .
Test intervals: .
Solution: , .
Section 5.5 – Real Zeros of Polynomial Functions
Finding real zeros of polynomials is essential for solving equations and analyzing graphs. The Rational Root Theorem and synthetic division are key tools for this process.
Rational Root Theorem: Possible rational zeros are .
Synthetic Division: Used to test possible zeros efficiently.
Example 5: Find a real root of .
Possible rational zeros: .
Test each using synthetic division to find actual zeros.
Section 5.6 – Complex Zeros & Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that every polynomial equation of degree n has exactly n roots (real or complex). Complex zeros often occur in conjugate pairs when coefficients are real.
Fundamental Theorem of Algebra (FTA): Every non-zero, single-variable polynomial of degree n has exactly n roots in the complex number system.
Complex Zeros: If a polynomial has real coefficients, non-real complex zeros occur in conjugate pairs.
Example 6: Solve .
Solution: .
