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
- Name degree, leading coefficient, terms, and polynomial type
- Use polynomial vocabulary and degree in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. What is the degree of p(x) = 3x^5 - 3x^2 + 4?
- The degree is the highest exponent on x that has a nonzero coefficient.
- The exponents present are 5, 2, and 0; the largest is 5.
- So the degree is 5.
Answer: 5
Practice problems
1. What is the degree of p(x) = 3x^5 - 3x^2 + 4?
Show solution
- The degree is the highest exponent on x that has a nonzero coefficient.
- The exponents present are 5, 2, and 0; the largest is 5.
- So the degree is 5.
Answer: 5
2. What is the degree of p(x) = 4x^2 + 4x^6 - 5x?
Show solution
- Degree is the largest exponent, not the exponent of the first term written.
- Scan every term: exponents are 2, 6, and 1.
- The largest is 6, so the degree is 6.
Answer: 6
3. What is the leading coefficient of p(x) = -5x^4 + 2x^2 - 6?
Show solution
- The leading coefficient is the coefficient of the highest-degree term.
- The highest-degree term is -5x^4.
- So the leading coefficient is -5.
Answer: -5
4. Written in standard form, what is the leading coefficient of p(x) = 3x - 7 + 2x^5?
Show solution
- First find the highest-degree term; its coefficient is the leading coefficient.
- The highest power is x^5, whose coefficient is 2.
- So the leading coefficient is 2.
Answer: 2
5. What is the constant term of p(x) = 3x^6 + 4x^2 - 3?
Show solution
- Core Practice: First identify exactly what the question is asking: What is the constant term of p(x) = 3x^6 + 4x^2 - 3?
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- The constant term is the term with no x (the x^0 term).
- Here that term is -3.
- So the constant term is -3.
- Check the result by substituting or estimating: the response should match -3 and make sense in the original problem.
Answer: -3
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