Polynomial Functions
A free College Algebra lesson from the “Advanced Function Types” unit, with a worked example and practice problems including step-by-step solutions.
Polynomial function graphs are guided by degree, leading coefficient, zeros, and multiplicity. Even multiplicity touches the x-axis, while odd multiplicity crosses.
What you'll learn
- Use end behavior
- Find zeros from factored form
- Interpret multiplicity
Worked example
Problem. For f(x) = (x - 2)^2(x + 3), which zero touches the x-axis?
- x - 2 gives zero 2.
- The factor is squared, so multiplicity is even.
- Even multiplicity touches.
Answer: 2
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. For f(x) = (x + 5)(x - 1), the zeros are...
Choices: -5 and 1 · 5 and -1 · 5 and 1 · -5 and -1
Show solution
- Set each factor equal to zero.
Answer: -5 and 1
2. A zero with multiplicity 2 usually...
Choices: Touches and turns · Crosses · Disappears · Is never graphed
Show solution
- Even multiplicity.
Answer: Touches and turns
3. The graph of y = -x^4 has ends that...
Choices: Both fall · Both rise · Left rises and right falls · Left falls and right rises
Show solution
- Even degree with negative leading coefficient.
Answer: Both fall
4. A degree 5 polynomial with positive leading coefficient has end behavior...
Choices: Left down, right up · Left up, right down · Both up · Both down
Show solution
- Odd degree with positive leading coefficient rises to the right.
Answer: Left down, right up
5. The zeros of f(x) = (x - 2)(x + 1) are...
Choices: 2 and -1 · -2 and 1 · 2 and 1 · -2 and -1
Show solution
- Set each factor equal to zero.
Answer: 2 and -1
Practice this interactively with instant feedback and an AI tutor.
Practice Polynomial Functions Take the free placement check