Rucete ✏ AP Precalculus In a Nutshell
2. Polynomial Functions
This chapter explores polynomial functions, including their structure, behavior, graphs, zeros, symmetry, and inequalities. You'll learn to analyze, factor, and sketch polynomial graphs using key theorems and properties essential for precalculus and beyond.
- Structure of Polynomial Functions
A polynomial is written as:
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ ≠ 0 and n is a non-negative integer.
Leading term: aₙxⁿ
Leading coefficient: aₙ
Constant term: a₀
A polynomial of:
- Degree 0 → constant
- Degree 1 → linear
- Degree 2 → quadratic
- Degree 3+ → cubic, quartic, etc.
Written in descending powers. Missing terms are assumed to have 0 coefficients.
- Rates of Change and Relative Extrema
Rate of change helps classify functions and concavity.
Relative extrema (local max/min): where function changes from increasing to decreasing or vice versa.
Extreme Value Theorem: Continuous polynomial on [a, b] has both a max and min.
Local Extrema Theorem: A polynomial of degree n has at most n − 1 relative extrema.
Between two real zeros, there must be at least one extremum.
- Absolute Extrema and Inflection Points
Absolute extrema: greatest and least values on a closed interval.
Inflection point: where concavity changes.
Point of Inflection Theorem: A degree-n polynomial has at most n − 2 inflection points.
- Zeros of Polynomial Functions
Zeros = values of x where f(x) = 0
- Found graphically (x-intercepts) or algebraically (factoring, quadratic formula, etc.)
Complex zeros: imaginary or real; total number (including repeats) = degree.
Conjugate pairs: If a + bi is a zero, then a − bi is also a zero.
Multiplicity: how many times a factor repeats.
- Even → graph is tangent to x-axis
- Odd → graph crosses x-axis
- Symmetry of Polynomial Functions
Even functions: f(−x) = f(x), symmetric about y-axis
- Example: f(x) = x²
Odd functions: f(−x) = −f(x), symmetric about origin
- Example: f(x) = x³
- End Behavior
Determined by leading term (degree and sign of leading coefficient):
| Degree | Coefficient | Left End | Right End |
|---|---|---|---|
| Even | Positive | ↑ | ↑ |
| Even | Negative | ↓ | ↓ |
| Odd | Positive | ↓ | ↑ |
| Odd | Negative | ↑ | ↓ |
Use limit notation:
- Example: lim(x→∞) f(x) = ∞, lim(x→−∞) f(x) = −∞
- Graphing Polynomial Functions (7-Step Strategy)
- State degree → max # of x-intercepts
- Find extrema and inflection point count
- Test end behavior
- Find x-intercepts (zeros)
- State multiplicity of each zero
- Find y-intercept (f(0))
- Sketch graph using all info above
- Binomial Theorem and Pascal’s Triangle
Expands (a + b)ⁿ without multiplying repeatedly.
Binomial coefficient: C(n, r) = n! / [r!(n − r)!]
Coefficients follow Pascal’s Triangle
Example: (x − 3y)⁴ = x⁴ − 12x³y + 54x²y² − 108xy³ + 81y⁴
- Polynomial Inequalities
Steps to solve:
- Rewrite in form f(x) < 0, f(x) > 0, etc.
- Find roots
- Plot roots on number line
- Test sign of f(x) in each interval
- Select intervals that satisfy inequality
Remember:
- Strict inequality (<, >) → open circles
- Inclusive (≤, ≥) → closed circles
In a Nutshell
Polynomial functions are foundational in algebra and precalculus. You’ve learned how to identify their structure, analyze their zeros, graph their behavior, and expand or solve expressions using key tools like the binomial theorem and limit notation. Understanding symmetry, end behavior, and inequalities prepares you for higher-level calculus and function analysis.