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
- Review polynomial vocabulary, graphs, zeros, division, and factor connections
- Choose the correct function, graph, or modeling tool from mixed prompts
- Explain why the selected method fits the problem
Worked example
Problem. What is the degree of p(x) = -3x^3 + 3x^2 - 5?
- Worked Example: First identify exactly what the question is asking: What is the degree of p(x) = -3x^3 + 3x^2 - 5?
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- The degree is the highest exponent with nonzero coefficient.
- The highest power shown is x^3.
- So the degree is 3.
- 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. Unit review 1 (Polynomial Vocabulary and Degree): What is the degree of p(x) = -3x^3 + 3x^2 - 5?
Choices: 3 · 2 · 5
Show solution
- Unit Review: First identify exactly what the question is asking: What is the degree of p(x) = -3x^3 + 3x^2 - 5?
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- The degree is the highest exponent with nonzero coefficient.
- The highest power shown is x^3.
- So the degree is 3.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 3
2. Unit review 2 (End Behavior): An even-degree polynomial with a positive leading coefficient has end behavior:
Choices: both ends up · both ends down · left down and right up · left up and right down
Show solution
- Even degree means the ends go the same direction.
- Positive leading coefficient makes the right end rise.
- Both ends rise.
Answer: both ends up
3. Unit review 3 (Zeros and Multiplicity): A zero with even multiplicity usually makes the graph:
Choices: touch and turn at the x-axis · cross sharply · create a vertical asymptote · remove the y-intercept
Show solution
- Unit Review: First identify exactly what the question is asking: A zero with even multiplicity usually makes the graph:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Even multiplicity keeps the sign the same.
- The graph touches and turns around.
- It does not cross at that zero.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: touch and turn at the x-axis
4. Unit review 4 (Graphing Polynomial Functions): A polynomial sketch should usually start with:
Choices: end behavior and zeros · only the y-axis label · a logarithm table · a denominator restriction
Show solution
- End behavior gives the broad direction.
- Zeros place x-axis interactions.
- Then multiplicity and points refine the sketch.
Answer: end behavior and zeros
5. Unit review 5 (Polynomial Division): Polynomial long division is needed when:
Choices: the divisor does not divide the dividend evenly · the dividend is a single constant · there are no variables · the divisor equals 1
Show solution
- If a factor cancels cleanly, simplify directly.
- Otherwise long division gives a quotient plus a remainder.
- The remainder has lower degree than the divisor.
Answer: the divisor does not divide the dividend evenly
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