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Understanding Polynomial Functions
AP Precalculus • Unit 1
Understanding Polynomial Functions
Build a strong conceptual foundation of polynomial functions — definitions, degrees, evaluation, and classification — essential for AP success and future calculus.
Lesson Objectives
- Define polynomial functions and their structure
- Identify degree and leading coefficient
- Evaluate polynomial functions
- Classify polynomials by degree
- Distinguish polynomial and non-polynomial expressions
1. What Is a Polynomial Function?
A polynomial function is a function that can be written in the form:
f(x) = anxn + an−1xn−1 + … + a1x + a0
- All exponents are whole numbers
- No variables in denominators or radicals
- Coefficients are real numbers
📌 Image placeholder: polynomial structure diagram
2. Polynomial vs Non-Polynomial
Example 1
f(x) = 4x³ − 2x + 7
✔ All exponents are non-negative integers → Polynomial
f(x) = 4x³ − 2x + 7
✔ All exponents are non-negative integers → Polynomial
Example 2
g(x) = √x + 3
✘ Fractional exponent → Not a polynomial
g(x) = √x + 3
✘ Fractional exponent → Not a polynomial
3. Degree and Leading Coefficient
Example 3
f(x) = −4x³ + 6x² − x + 9
f(x) = −4x³ + 6x² − x + 9
- Degree = 3
- Leading coefficient = −4
4. Evaluating Polynomial Functions
Example 4
f(x) = 2x³ − 5x + 1
f(2) = 2(8) − 10 + 1 = 7
f(x) = 2x³ − 5x + 1
f(2) = 2(8) − 10 + 1 = 7
Example 5
f(x) = x² − 3x + 4
f(a) = a² − 3a + 4
f(x) = x² − 3x + 4
f(a) = a² − 3a + 4
Example 6 (AP Style)
f(x) = x² + 2x
f(a + h) = (a + h)² + 2(a + h)
f(x) = x² + 2x
f(a + h) = (a + h)² + 2(a + h)
5. Types of Polynomial Functions
| Degree | Name | General Form | Graph Shape |
|---|---|---|---|
| 0 | Constant | f(x)=c | Horizontal line |
| 1 | Linear | mx+b | Straight line |
| 2 | Quadratic | ax²+bx+c | Parabola |
| 3 | Cubic | ax³+bx²+cx+d | S-curve |
| 4+ | Higher degree | — | Multiple turns |
📊 Image placeholder: comparison of polynomial graphs
6. Classifying Polynomials
Example 7
- 3x − 7 → Linear
- x² − 9 → Quadratic
- x⁴ + x² → Degree 4 polynomial
Practice Questions
- Find the degree of f(x)=6x⁵−2x+1
- Evaluate f(−2) for f(x)=x³−4x
- Is f(x)=√(x²+1) a polynomial? Explain.
✅ Show Answer Key
- Degree = 5
- f(−2) = −8 + 8 = 0
- No — square root creates a fractional exponent.
© Aviate Learning – AP Precalculus
🎯 Interactive: Build Your Own Polynomial
🔍 Use sliders to change coefficients and observe how the degree and shape of the polynomial change.
📈 Interactive: Polynomial End Behavior
🔁 Toggle between even and odd degrees and flip the leading coefficient to explore end behavior.
❌ Interactive: Zeros & Multiplicity
✨ Observe how roots with different multiplicities affect the graph’s behavior at the x-axis.
🔍 Interactive: Comparing Polynomial Degrees
📊 Compare how linear, quadratic, and cubic polynomials behave on the same axes.