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Polynomial Vocabulary and Degree

A free Precalculus lesson from the “Polynomial Functions” unit, with a worked example and practice problems including step-by-step solutions.

Degree and leading coefficient control the broad behavior of a polynomial. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Polynomial models describe smooth turning behavior in design, approximation, data fitting, and STEM modeling.

Worked example

Problem. What is the degree of p(x) = -3x^3 + 3x^2 - 5?

  1. Worked Example: First identify exactly what the question is asking: What is the degree of p(x) = -3x^3 + 3x^2 - 5?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The degree is the highest exponent with nonzero coefficient.
  4. The highest power shown is x^3.
  5. So the degree is 3.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 3

Practice problems

1. What is the degree of p(x) = -3x^3 + 3x^2 - 5?

Choices: 3 · 2 · 5

Show solution
  1. Warm-up: First identify exactly what the question is asking: What is the degree of p(x) = -3x^3 + 3x^2 - 5?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The degree is the highest exponent with nonzero coefficient.
  4. The highest power shown is x^3.
  5. So the degree is 3.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 3

2. What is the degree of p(x) = 4x^4 + 3x^2 - 5?

Choices: 4 · 3 · 2 · 5

Show solution
  1. Warm-up: First identify exactly what the question is asking: What is the degree of p(x) = 4x^4 + 3x^2 - 5?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The degree is the highest exponent with nonzero coefficient.
  4. The highest power shown is x^4.
  5. So the degree is 4.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 4

3. What is the degree of p(x) = -5x^5 + 3x^2 - 5?

Choices: 5 · 4 · 2

Show solution
  1. Core Practice: First identify exactly what the question is asking: What is the degree of p(x) = -5x^5 + 3x^2 - 5?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The degree is the highest exponent with nonzero coefficient.
  4. The highest power shown is x^5.
  5. So the degree is 5.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 5

4. What is the degree of p(x) = 6x^6 + 3x^2 - 5?

Choices: 6 · 5 · 2

Show solution
  1. Core Practice: First identify exactly what the question is asking: What is the degree of p(x) = 6x^6 + 3x^2 - 5?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The degree is the highest exponent with nonzero coefficient.
  4. The highest power shown is x^6.
  5. So the degree is 6.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 6

5. What is the degree of p(x) = -2x^2 + 3x^2 - 5?

Choices: 2 · 1 · 5

Show solution
  1. Core Practice: First identify exactly what the question is asking: What is the degree of p(x) = -2x^2 + 3x^2 - 5?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The degree is the highest exponent with nonzero coefficient.
  4. The highest power shown is x^2.
  5. So the degree is 2.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 2

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