CMClearMathAcademy

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 with zeros 2 and -3. Enter factored form.

  1. Worked Example: First identify exactly what the question is asking: Build a monic polynomial with zeros 2 and -3. Enter 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. Zero 2 gives factor x - 2.
  4. Zero -3 gives factor x + 3.
  5. A monic polynomial uses leading coefficient 1.
  6. Check the result by substituting or estimating: the response should match (x - 2)(x + 3) and make sense in the original problem.

Answer: (x - 2)(x + 3)

Practice problems

1. Build a monic polynomial with zeros 2 and -3. Enter factored form.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Build a monic polynomial with zeros 2 and -3. Enter 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. Zero 2 gives factor x - 2.
  4. Zero -3 gives factor x + 3.
  5. A monic polynomial uses leading coefficient 1.
  6. Check the result by substituting or estimating: the response should match (x - 2)(x + 3) and make sense in the original problem.

Answer: (x - 2)(x + 3)

2. Build a monic polynomial with zeros 3 and -4. Enter factored form.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Build a monic polynomial with zeros 3 and -4. Enter 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. Zero 3 gives factor x - 3.
  4. Zero -4 gives factor x + 4.
  5. A monic polynomial uses leading coefficient 1.
  6. Check the result by substituting or estimating: the response should match (x - 3)(x + 4) and make sense in the original problem.

Answer: (x - 3)(x + 4)

3. Build a monic polynomial with zeros 4 and -2. Enter factored form.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Build a monic polynomial with zeros 4 and -2. Enter 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. Zero 4 gives factor x - 4.
  4. Zero -2 gives factor x + 2.
  5. A monic polynomial uses leading coefficient 1.
  6. Check the result by substituting or estimating: the response should match (x - 4)(x + 2) and make sense in the original problem.

Answer: (x - 4)(x + 2)

4. Build a polynomial with leading coefficient 2 and zeros 1 and 3. Enter factored form.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Build a polynomial with leading coefficient 2 and zeros 1 and 3. Enter 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 become factors.
  4. Include the leading coefficient 2.
  5. The factored polynomial is 2(x - 1)(x - 3).
  6. Check the result by substituting or estimating: the response should match 2(x - 1)(x - 3) and make sense in the original problem.

Answer: 2(x - 1)(x - 3)

5. Build a polynomial with leading coefficient 2 and zeros 1 and 4. Enter factored form.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Build a polynomial with leading coefficient 2 and zeros 1 and 4. Enter 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 become factors.
  4. Include the leading coefficient 2.
  5. The factored polynomial is 2(x - 1)(x - 4).
  6. Check the result by substituting or estimating: the response should match 2(x - 1)(x - 4) and make sense in the original problem.

Answer: 2(x - 1)(x - 4)

Practice this interactively with instant feedback and an AI tutor.

Practice Building Polynomials from Zeros Take the free placement check

More Precalculus lessons