Module 3 Study Guide
Polynomial & Rational Functions - Comprehensive Review
Module 3: Polynomial & Rational Functions
Comprehensive Study Guide
1. Polynomial Functions
Definition
A polynomial function has the form:
f(x) = anxn + an-1xn-1 + ... + a1x + a0
Degree: Highest power of x (n)
Leading Coefficient: an (coefficient of highest degree term)
Standard Form: Terms arranged in descending order of degree
End Behavior (Leading Term Test)
| Degree | Leading Coeff. | As x → -∞ | As x → +∞ |
|---|---|---|---|
| Even | Positive | f(x) → +∞ | f(x) → +∞ |
| Even | Negative | f(x) → -∞ | f(x) → -∞ |
| Odd | Positive | f(x) → -∞ | f(x) → +∞ |
| Odd | Negative | f(x) → +∞ | f(x) → -∞ |
Zeros and Multiplicity
Zero: Value where f(x) = 0
Multiplicity: Number of times a zero appears as a factor
- Odd multiplicity (1, 3, 5...): Graph CROSSES x-axis
- Even multiplicity (2, 4, 6...): Graph TOUCHES x-axis (bounces)
Example
f(x) = (x + 2)2(x - 1)3
Zeros: x = -2 (mult. 2, touches), x = 1 (mult. 3, crosses)
Degree: 5 (odd), Leading coeff: 1 (positive)
End behavior: Left down, right up
2. Factoring Polynomials
Common Factoring Patterns
Difference of Squares: a2 - b2 = (a + b)(a - b)
Sum of Cubes: a3 + b3 = (a + b)(a2 - ab + b2)
Difference of Cubes: a3 - b3 = (a - b)(a2 + ab + b2)
Perfect Square Trinomial: a2 ± 2ab + b2 = (a ± b)2
Memory Tip for Cubes: "Same, Opposite, Always Positive"
- First binomial: SAME sign as original
- Middle term in trinomial: OPPOSITE sign
- Last term in trinomial: ALWAYS positive
Rational Root Theorem
For polynomial f(x) = anxn + ... + a0 with integer coefficients:
If p/q is a rational zero (in lowest terms), then:
- p is a factor of the constant term a0
- q is a factor of the leading coefficient an
Synthetic Division
Quick method to divide polynomial by (x - c)
- If remainder = 0, then c is a zero
- Quotient gives reduced polynomial
- Continue factoring the quotient
Example: Factor Completely
f(x) = x3 - 7x + 6
Step 1: Potential zeros: ±1, ±2, ±3, ±6
Step 2: Test x = 1: f(1) = 1 - 7 + 6 = 0
Step 3: Synthetic division → quotient: x2 + x - 6
Step 4: Factor: x2 + x - 6 = (x + 3)(x - 2)
Answer: f(x) = (x - 1)(x + 3)(x - 2)
3. Rational Functions
Definition
f(x) = P(x) / Q(x) where P and Q are polynomials, Q(x) ≠ 0
Domain: All real numbers except where Q(x) = 0
Vertical Asymptotes
Definition: Line x = a where function approaches ±∞
How to Find:
- Factor numerator and denominator
- Cancel common factors
- Set REMAINING denominator = 0
- Solve for x
Horizontal Asymptotes
| Condition | Horizontal Asymptote |
|---|---|
| deg(P) < deg(Q) | y = 0 |
| deg(P) = deg(Q) | y = (leading coeff of P)/(leading coeff of Q) |
| deg(P) > deg(Q) | No horizontal asymptote |
Holes (Removable Discontinuities)
Occur when: Factor cancels from numerator AND denominator
To find hole location:
- Set cancelled factor = 0, solve for x
- Substitute x into SIMPLIFIED function for y-coordinate
- Hole at point (x, y)
Example: Complete Analysis
f(x) = (x2 - 4)/(x2 + x - 6)
Factor: (x-2)(x+2) / [(x+3)(x-2)]
Hole: x = 2, y = 4/5 → (2, 4/5)
Vertical asymptote: x = -3
Horizontal asymptote: y = 1 (degrees equal, coefficients 1/1)
X-intercept: x = -2
Y-intercept: f(0) = -4/-6 = 2/3
4. Polynomial & Rational Inequalities
Test Interval Method (Polynomial Inequalities)
- Move all terms to one side (= 0 on other side)
- Factor completely
- Find all zeros (critical values)
- Plot zeros on number line
- Test a point in each interval
- Select intervals where inequality is true
- Check if boundaries included (≤ or ≥) or excluded (< or >)
Example: Polynomial Inequality
Solve: x2 - 5x + 6 < 0
Factor: (x - 2)(x - 3) < 0
Zeros: x = 2, 3
Test intervals:
(-∞, 2): Test x=0 → (+) Positive
(2, 3): Test x=2.5 → (-) Negative
(3, ∞): Test x=4 → (+) Positive
Solution: (2, 3) or 2 < x < 3
Rational Inequalities
Critical Values: Zeros of numerator AND denominator
NEVER multiply both sides by denominator!
Sign could change. Use test interval method instead.
NEVER include values where denominator = 0!
- Move all terms to one side
- Combine into single rational expression
- Factor numerator and denominator
- Find critical values (num = 0 AND den = 0)
- Test intervals
- Select appropriate intervals
- Include numerator zeros for ≤ or ≥ (if denominator ≠ 0)
- NEVER include denominator zeros
Example: Rational Inequality
Solve: (x - 3)/(x + 1) ≥ 0
Critical values: x = 3 (num), x = -1 (den)
Test intervals:
(-∞, -1): (-)/(-) = +
(-1, 3): (-)/(+) = -
(3, ∞): (+)/(+) = +
Include x = 3: Yes (≥ and den ≠ 0)
Include x = -1: NO (denominator = 0)
Solution: (-∞, -1) ∪ [3, ∞)
5. Common Mistakes to Avoid
1. End Behavior
Don't just look at leading coefficient - check BOTH degree (odd/even) AND sign!
2. Multiplicity
Even multiplicity → touches (bounces), Odd multiplicity → crosses. Don't mix these up!
3. Holes vs. Vertical Asymptotes
Cancel common factors FIRST! Cancelled factors create holes, not vertical asymptotes.
4. Horizontal Asymptotes
Compare degrees AFTER simplifying. If deg(num) > deg(den), NO horizontal asymptote!
5. Inequality Boundaries
< or > → exclude boundaries (open intervals)
≤ or ≥ → include boundaries (closed intervals)
ALWAYS exclude values where denominator = 0!
6. Sum/Difference of Cubes
Remember: "Same, Opposite, Always Positive"
a3 + b3 = (a + b)(a2 - ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)
6. Quick Practice Problems
1. Find end behavior of f(x) = -3x6 + 2x3 - 5
Answer: Both ends down (even degree, negative leading coeff)
2. Factor: x3 - 27
Answer: (x - 3)(x2 + 3x + 9)
3. Find horizontal asymptote of f(x) = (4x2 - 1)/(2x2 + 3)
Answer: y = 4/2 = 2 (degrees equal)
4. Solve: (x + 2)(x - 4) > 0
Answer: x < -2 or x > 4
5. Where is the hole in g(x) = (x2 - 1)/(x2 - 2x + 1)?
Answer: (x-1)(x+1)/(x-1)2 → hole at (1, 2)
Module 3 Summary
- Polynomials: Degree, leading coefficient determine end behavior. Zeros and multiplicity determine x-intercepts and crossing behavior.
- Factoring: Use GCF, special patterns (difference of squares, sum/difference of cubes), Rational Root Theorem, and synthetic division.
- Rational Functions: Find domain, vertical asymptotes (remaining denominator zeros), horizontal asymptotes (compare degrees), and holes (cancelled factors).
- Inequalities: Test intervals, never multiply by denominator, never include denominator zeros, check boundary inclusion.