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Unit 4 Review and Quiz

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

This checkpoint confirms full polynomial-function behavior before rational functions. 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^5 - 3x^2 + 4?

  1. The degree is the highest exponent on x that has a nonzero coefficient.
  2. The exponents present are 5, 2, and 0; the largest is 5.
  3. So the degree is 5.

Answer: 5

Practice problems

1. Unit review 1 (Polynomial Vocabulary and Degree): What is the degree of p(x) = 3x^5 - 3x^2 + 4?

Show solution
  1. The degree is the highest exponent on x that has a nonzero coefficient.
  2. The exponents present are 5, 2, and 0; the largest is 5.
  3. So the degree is 5.

Answer: 5

2. Unit review 2 (End Behavior): Which phrase describes the ends of p(x) = -4x^6 + 3?

Choices: both ends down · both ends up · left end down, right end up · left end up, right end down

Show solution
  1. The degree 6 is even, so both ends point the same direction.
  2. The leading coefficient -4 is negative, so the right end falls.
  3. Same direction and right end down means both ends go down.

Answer: both ends down

3. Unit review 3 (Zeros and Multiplicity): For p(x) = (x - 5)^2(x - 6)^2(x - 12)^4, the degree equals the sum of all multiplicities. Find the degree.

Show solution
  1. Unit Review: First identify exactly what the question is asking: For p(x) = (x - 5)^2(x - 6)^2(x - 12)^4, the degree equals the sum of all multiplicities. Find the degree.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Add the exponents: 2 + 2 + 4.
  4. 2 + 2 + 4 = 8.
  5. The sum of multiplicities is the degree, so the degree is 8.
  6. Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.

Answer: 8

4. Unit review 4 (Graphing Polynomial Functions): A polynomial has degree 7. What is the maximum number of turning points its graph can have?

Show solution
  1. Unit Review: First identify exactly what the question is asking: A polynomial has degree 7. What is the maximum number of turning points its graph can have?
  2. For polynomial work, use degree, leading terms, and like terms to keep the expression organized.
  3. A degree-n graph turns at most n - 1 times.
  4. Here n = 7, so the maximum is 7 - 1.
  5. That is 6 turning points.
  6. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

5. Unit review 5 (Polynomial Division): By the Remainder Theorem, find the remainder when p(x) = x^2 + 2 is divided by (x + 3).

Show solution
  1. Unit Review: First identify exactly what the question is asking: By the Remainder Theorem, find the remainder when p(x) = x^2 + 2 is divided by (x + 3).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Dividing by (x + 3) means evaluating p at x = -3.
  4. p(-3) = (-3)^2 + 2 = 9 + 2, since (-3)^2 = 9.
  5. So the remainder is 11.
  6. Check the result by substituting or estimating: the response should match 11 and make sense in the original problem.

Answer: 11

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