Polynomial and Rational Functions: Study Notes for College Algebra

Polynomial and Rational Functions

Section 3.3: Polynomial Functions

Polynomial functions are algebraic expressions involving sums of powers of x with real coefficients. Understanding their structure, degree, and transformations is essential in College Algebra.

• Parent Function: The simplest form of a polynomial in a given family (e.g., quadratic parent function is $f(x) = x^2$).

• Degree: The highest power of x in the polynomial. Determines the end behavior and number of possible real zeros.

• Transformations: Shifts, reflections, and stretches/compressions applied to the parent function to obtain the given function.

Example:

• Given $f(x) = (x + 2)^2 - 3$:

  • Parent function: $f(x) = x^2$

  • Degree: 2

  • Transformation: Shift 2 units left, then 3 units down

Finding a Polynomial with Given Zeros

To construct a polynomial with specified zeros, use the fact that if r is a zero, then $(x - r)$ is a factor.

• Example: Zeros at 2, 3, 5: $f(x) = a(x-2)(x-3)(x-5)$, where a is a leading coefficient (often 1 unless otherwise specified).

Graphing and Analyzing Polynomial Functions

• Multiplicity: The number of times a zero is repeated. If a zero has even multiplicity, the graph touches the x-axis and turns around; if odd, it crosses the axis.

• End Behavior: Determined by the leading term. For $f(x) = ax^n$:

  • If n is even and a positive: both ends up.

  • If n is even and a negative: both ends down.

  • If n is odd and a positive: left down, right up.

  • If n is odd and a negative: left up, right down.

• Example: $f(x) = x^3 - 2x^2 + x$ has degree 3 (odd), leading coefficient 1 (positive), so left end down, right end up.

Section 3.4: Rational Zeros Theorem

The Rational Zeros Theorem helps find all possible rational zeros of a polynomial function with integer coefficients.

• Theorem: If $f(x) = a_nx^n + ... + a_0$ has integer coefficients, any rational zero $\frac{p}{q}$ must have p dividing $a_0$ and q dividing $a_n$.

• Steps:

  1. List all factors of the constant term ($a_0$) and leading coefficient ($a_n$).

  2. Form all possible $\pm\frac{p}{q}$ combinations.

  3. Test each candidate in the polynomial.

• Example: For $f(x) = 2x^3 - 3x^2 - 8x + 3$, possible rational zeros are $\pm1, \pm3, \pm\frac{1}{2}, \pm\frac{3}{2}$.

Section 3.5: Rational Functions

Rational functions are quotients of two polynomials. Their properties depend on the degrees and factors of numerator and denominator.

• Domain: All real numbers except where the denominator is zero.

• Vertical Asymptotes: Values of x where the denominator is zero (and numerator is not zero at that point).

• Horizontal Asymptotes: Determined by comparing degrees of numerator and denominator:

  • If degree numerator < degree denominator: $y = 0$

  • If degrees equal: $y = \frac{\text{leading coefficient numerator}}{\text{leading coefficient denominator}}$

  • If degree numerator > degree denominator: No horizontal asymptote (may have an oblique/slant asymptote)

• x-intercepts: Set numerator equal to zero (and denominator not zero).

• y-intercept: Evaluate at $x = 0$.

Graphing Rational Functions

• Identify asymptotes, intercepts, and holes (if any).

• Plot key points and sketch the general shape, considering end behavior and behavior near asymptotes.

• Example: $f(x) = \frac{2x-1}{x+3}$

  • Vertical asymptote: $x = -3$

  • Horizontal asymptote: $y = 2$

  • x-intercept: $x = \frac{1}{2}$

  • y-intercept: $y = -\frac{1}{3}$

Summary Table: Properties of Rational Functions

Property	Description	How to Find
Domain	All real numbers except where denominator = 0	Solve denominator = 0
Vertical Asymptote	Where denominator = 0 (numerator ≠ 0)	Solve denominator = 0
Horizontal Asymptote	End behavior as x → ±∞	Compare degrees of numerator and denominator
x-intercept	Where graph crosses x-axis	Set numerator = 0
y-intercept	Where graph crosses y-axis	Set x = 0

Additional info:

• Some problems ask to match graphs to equations, requiring analysis of zeros, multiplicity, and end behavior.

• Practice includes constructing polynomials from zeros, applying transformations, and analyzing rational functions for intercepts and asymptotes.