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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

Why it matters: Long-term forecasts, stress curves, and break-even models often use polynomial functions. Degree and leading term predict far-end behavior, while zeros and multiplicity explain where the graph crosses or turns.

Worked example

Problem. For f(x) = (x - 2)^2(x + 3), which zero touches the x-axis?

  1. x - 2 gives zero 2.
  2. The factor is squared, so multiplicity is even.
  3. Even multiplicity touches.

Answer: 2

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
  1. Warm-up: First identify exactly what the question is asking: For f(x) = (x + 5)(x - 1), the zeros are...
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Set each factor equal to zero.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: -5 and 1

2. A zero with multiplicity 2 usually...

Choices: Touches and turns · Crosses · Disappears · Is never graphed

Show solution
  1. Core Practice: First identify exactly what the question is asking: A zero with multiplicity 2 usually...
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Even multiplicity.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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
  1. Challenge: First identify exactly what the question is asking: The graph of y = -x^4 has ends that...
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Even degree with negative leading coefficient.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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
  1. End Behavior: First identify exactly what the question is asking: A degree 5 polynomial with positive leading coefficient has end behavior...
  2. For polynomial work, use degree, leading terms, and like terms to keep the expression organized.
  3. Odd degree with positive leading coefficient rises to the right.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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
  1. Zeros: First identify exactly what the question is asking: The zeros of f(x) = (x - 2)(x + 1) are...
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Set each factor equal to zero.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 2 and -1

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