Polynomial Functions ✏ AP Precalculus

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
• Degree 0 → constant, Degree 1 → linear, Degree 2 → quadratic, Degree 3+ → cubic, quartic, etc.
• Written in descending powers. Missing terms = 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: Degree n → 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: Degree n → 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: real or imaginary; total = degree (including multiplicity)
• Conjugate pairs: If a + bi is a zero, then a − bi also is
• Multiplicity:
  • 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 (e.g., f(x) = x²)
• Odd functions: f(−x) = −f(x) → symmetric about origin (e.g., f(x) = x³)

- End Behavior

Determined by degree and sign of leading coefficient:

Degree	Coefficient	Left End	Right End
Even	Positive	↑	↑
Even	Negative	↓	↓
Odd	Positive	↓	↑
Odd	Negative	↑	↓

• Use limit notation: lim(x→∞) f(x) = ∞, lim(x→−∞) f(x) = −∞

- Graphing Polynomial Functions (7-Step Strategy)

1. State degree → max # of x-intercepts
2. Find extrema and inflection point count
3. Test end behavior
4. Find x-intercepts (zeros)
5. State multiplicity of each zero
6. Find y-intercept (f(0))
7. 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:
  1. Rewrite in form f(x) < 0, f(x) > 0, etc.
  2. Find roots
  3. Plot roots on number line
  4. Test sign of f(x) in each interval
  5. 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.