Module 3 Quick Reference Card
Polynomial & Rational Functions
1. POLYNOMIAL FUNCTIONS
Standard Form
f(x) = anxn + an-1xn-1 + ... + a0
Degree: n (highest power)
Leading Coefficient: an
Multiplicity
- Odd: Graph CROSSES x-axis
- Even: Graph TOUCHES x-axis
End Behavior (Leading Term Test)
| Degree | Leading Coeff | Left End | Right End |
|---|---|---|---|
| Even | Positive | Up (+∞) | Up (+∞) |
| Even | Negative | Down (-∞) | Down (-∞) |
| Odd | Positive | Down (-∞) | Up (+∞) |
| Odd | Negative | Up (+∞) | Down (-∞) |
2. FACTORING FORMULAS
Special Patterns
a2 - b2 = (a+b)(a-b)
a2 ± 2ab + b2 = (a±b)2
Cubes (Same, Opposite, Always +)
a3 + b3 = (a+b)(a2-ab+b2)
a3 - b3 = (a-b)(a2+ab+b2)
Rational Root Theorem
For f(x) = anxn + ... + a0, if p/q is a rational zero:
- p divides a0 (constant term)
- q divides an (leading coefficient)
3. RATIONAL FUNCTIONS
Definition: f(x) = P(x)/Q(x)
Vertical Asymptotes
- Factor numerator & denominator
- Cancel common factors
- Set remaining denominator = 0
Holes
Occur when factors cancel
- Find x: Set cancelled factor = 0
- Find y: Use simplified function
Horizontal Asymptotes
| Condition | Horizontal Asymptote |
|---|---|
| deg(P) < deg(Q) | y = 0 |
| deg(P) = deg(Q) | y = an/bn (ratio of leading coefficients) |
| deg(P) > deg(Q) | No horizontal asymptote |
4. SOLVING INEQUALITIES
Polynomial Inequalities
- Move all terms to one side
- Factor completely
- Find zeros (critical values)
- Test intervals
- Select appropriate regions
- Include/exclude boundaries
Rational Inequalities
- Move to one side
- Factor numerator & denominator
- Critical values: num AND den zeros
- Test intervals
- Select appropriate regions
- NEVER include den = 0!
Boundary Points
- < or >: Exclude boundary (open interval)
- ≤ or ≥: Include boundary (closed interval)
- ALWAYS exclude where denominator = 0
5. QUICK EXAMPLES
End Behavior
f(x) = -2x5 + 3x2 - 1
Odd degree, negative coeff
Answer: Left up, right down
Factor
x3 + 64 = x3 + 43
Answer: (x+4)(x2-4x+16)
Horizontal Asymptote
f(x) = (3x2-1)/(6x2+5)
Degrees equal: 2 = 2
Answer: y = 3/6 = 1/2
Solve Inequality
(x-1)(x+3) > 0
Zeros: -3, 1; Test intervals
Answer: x < -3 or x > 1
6. KEY REMINDERS
- Multiplicity: Even = touch, Odd = cross
- Cubes: Same sign, Opposite middle, Always positive last
- Holes vs VA: Cancel factors first! Cancelled = hole, Remaining = VA
- Never multiply inequality by denominator (sign could change)
- Rational Root Theorem: p/q where p|a0, q|an
- Horizontal asymptote: Compare degrees AFTER simplifying
Module 3: Polynomial & Rational Functions
College Algebra | Safaa Dabagh | 2026