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Polynomial Operations

A free Algebra I lesson from the “Exponents and Polynomials” unit, with a worked example and practice problems including step-by-step solutions.

Polynomial operations extend expression skills. Add and subtract by combining like terms; multiply by distributing each factor across the terms it touches.

What you'll learn

Why it matters: Area models, packaging layouts, and design measurements use polynomial multiplication when side lengths include unknown pieces.

Worked example

Problem. Simplify (2x + 3)(x + 4).

  1. Distribute 2x to get 2x^2 + 8x.
  2. Distribute 3 to get 3x + 12.
  3. Combine like terms: 2x^2 + 11x + 12.

Answer: 2x^2 + 11x + 12

Practice problems

1. Simplify (3x + 2) + (4x + 5).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Simplify (3x + 2) + (4x + 5).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Combine x terms and constants.
  4. 3x + 4x = 7x and 2 + 5 = 7.
  5. Check the result by substituting or estimating: the response should match 7x + 7 and make sense in the original problem.

Answer: 7x + 7

2. Simplify (6x - 1) - (2x + 3).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Simplify (6x - 1) - (2x + 3).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Distribute the subtraction.
  4. 6x - 1 - 2x - 3 = 4x - 4.
  5. Check the result by substituting or estimating: the response should match 4x - 4 and make sense in the original problem.

Answer: 4x - 4

3. Simplify 3x(2x + 5).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Simplify 3x(2x + 5).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Distribute 3x.
  4. 3x times 2x is 6x^2 and 3x times 5 is 15x.
  5. Check the result by substituting or estimating: the response should match 6x^2 + 15x and make sense in the original problem.

Answer: 6x^2 + 15x

4. Simplify (x^2 + 3x + 1) + (2x^2 - x + 4).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Simplify (x^2 + 3x + 1) + (2x^2 - x + 4).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Combine like terms.
  4. x^2 + 2x^2 = 3x^2, 3x - x = 2x, and 1 + 4 = 5.
  5. Check the result by substituting or estimating: the response should match 3x^2 + 2x + 5 and make sense in the original problem.

Answer: 3x^2 + 2x + 5

5. Simplify (5x^2 + x - 6) - (2x^2 - 3x + 1).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Simplify (5x^2 + x - 6) - (2x^2 - 3x + 1).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Change signs in the second polynomial.
  4. Combine to get 3x^2 + 4x - 7.
  5. Check the result by substituting or estimating: the response should match 3x^2 + 4x - 7 and make sense in the original problem.

Answer: 3x^2 + 4x - 7

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