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Building Polynomials from Zeros

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

Zeros become factors. Multiplicity tells how many times each factor appears. 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. Build a monic polynomial whose zeros are 6 and 11. Enter the factored form.

  1. A monic polynomial has leading coefficient 1.
  2. Zero 6 gives factor x - 6; zero 11 gives factor x - 11.
  3. Multiply the factors: (x - 6)(x - 11).

Answer: (x - 6)(x - 11)

Practice problems

1. Build a monic polynomial whose zeros are 6 and 11. Enter the factored form.

Show solution
  1. A monic polynomial has leading coefficient 1.
  2. Zero 6 gives factor x - 6; zero 11 gives factor x - 11.
  3. Multiply the factors: (x - 6)(x - 11).

Answer: (x - 6)(x - 11)

2. Build a monic polynomial with one positive zero 7 and one negative zero -12. Enter the factored form.

Show solution
  1. Positive zero 7 gives factor x - 7.
  2. Negative zero -12 gives factor x + 12, since x - (-12) = x + 12.
  3. The monic product is (x - 7)(x + 12).

Answer: (x - 7)(x + 12)

3. A polynomial has a zero at x = 8. Which factor must it contain?

Choices: x - 8 · x + 8 · 8x · 8x - 1

Show solution
  1. Core Practice: First identify exactly what the question is asking: A polynomial has a zero at x = 8. Which factor must it contain?
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. A zero at x = c means p(8) = 0.
  4. That happens exactly when (x - 8) divides p(x).
  5. So the factor is x - 8.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x - 8

4. A polynomial has a NEGATIVE zero at x = -10. Which factor must it contain?

Choices: x + 10 · x - 10 · 10x · 10x + 1

Show solution
  1. Core Practice: First identify exactly what the question is asking: A polynomial has a NEGATIVE zero at x = -10. Which factor must it contain?
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. A zero at x = -10 means p(-10) = 0.
  4. The matching factor is x - (-10).
  5. That simplifies to x + 10.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x + 10

5. Build a polynomial with leading coefficient 4 and zeros 5 and 11. Enter the factored form.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Build a polynomial with leading coefficient 4 and zeros 5 and 11. Enter the factored form.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Zeros 5 and 11 give factors x - 5 and x - 11.
  4. Multiply by the leading coefficient 4.
  5. The result is 4(x - 5)(x - 11).
  6. Check the result by substituting or estimating: the response should match 4(x - 5)(x - 11) and make sense in the original problem.

Answer: 4(x - 5)(x - 11)

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