Nonlinear Functions • Topic 3 of 3

Polynomial Functions

A polynomial function is a sum of terms of the form (coefficient)·xⁿ with whole-number exponents, such as f(x) = 2x³ − 4x + 1. The degree is the highest power of x, and it shapes the graph’s overall behavior and how many turns or roots it can have. Linear (degree 1) and quadratic (degree 2) are the simplest polynomials; cubics are degree 3. To evaluate, substitute and apply each power carefully, watching signs when the input is negative — for f(x) = x³ + 2, f(−2) = (−2)³ + 2 = −8 + 2 = −6. The SAT tests reading the degree and evaluating low-degree polynomials accurately.

sat30t3 graphOPolynomial functiony = x³ — a cubic with an inflection at the origin

✅ Solved examples

1. What is the degree of f(x) = 2x³ + 5x − 1?
The highest power is 3.
2. If f(x) = x³ + 2, find f(2).
8 + 2 = 10.
3. If f(x) = x³ − 1, find f(−2).
(−2)³ − 1 = −8 − 1 = −9.
4. What is the degree of f(x) = x⁴ + 3x² + 7?
4.

✏️ Practice — try these, take hints as needed

1. What is the degree of f(x) = 5x³ + 2x − 8?
Highest power.
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3.
2. If f(x) = x³ + 4, find f(2).
8 + 4.
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12.
3. If f(x) = 2x³, find f(3).
2 · 27.
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54.
4. If f(x) = x³ + 1, find f(−1).
(−1)³ + 1.
−1 + 1.
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0.
5. What is the degree of f(x) = 3x⁵ + x² − 4?
Highest power.
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5.

📝 Topic test — 8 questions

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