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Polynomial Arithmetic

A free College Algebra lesson from the “Polynomial Arithmetic and Theorems” unit, with a worked example and practice problems including step-by-step solutions.

Polynomial operations follow the structure of terms, coefficients, variables, and exponents. To add or subtract polynomials, combine like terms; to multiply, distribute each term and then combine like terms. This matters because polynomial expressions model area, volume, rates, and higher-degree functions. When practicing, organize terms by degree and watch signs carefully, especially when subtracting an entire polynomial. A common mistake is combining unlike terms such as x^2 and x, or forgetting to distribute a negative sign to every term in parentheses.

What you'll learn

Why it matters: Area formulas, physics expressions, cost models, and higher-degree functions often combine polynomial pieces. Organizing by degree makes the expression readable and prevents unlike terms from being merged.

Worked example

Problem. Simplify (3x^2 + 4x - 1) + (2x^2 - 9x + 5).

  1. Combine x^2 terms.
  2. Combine x terms.
  3. Combine constants.

Answer: 5x^2 - 5x + 4

Practice problems

1. Simplify (2x + 7) + (5x - 3).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Simplify (2x + 7) + (5x - 3).
  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. Check the result by substituting or estimating: the response should match 7x + 4 and make sense in the original problem.

Answer: 7x + 4

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

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

Answer: 4x^2 - 6x

3. The degree of 9x^5 - 3x^2 + 1 is...

Choices: 5 · 9 · 2 · 1

Show solution
  1. Challenge: First identify exactly what the question is asking: The degree of 9x^5 - 3x^2 + 1 is...
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Degree is the highest exponent.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 5

4. Add (3x^2 + 2x - 1) + (x^2 - 5x + 7).

Choices: 4x^2 - 3x + 6 · 4x^2 + 7x + 6 · 3x^4 - 3x + 6 · 2x^2 - 3x - 8

Show solution
  1. Addition: First identify exactly what the question is asking: Add (3x^2 + 2x - 1) + (x^2 - 5x + 7).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Combine like terms.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 4x^2 - 3x + 6

5. Subtract (5x^2 - 4) - (2x^2 + 3x - 9).

Choices: 3x^2 - 3x + 5 · 3x^2 + 3x - 13 · 7x^2 + 3x - 13 · 3x^2 - 3x - 5

Show solution
  1. Subtraction: First identify exactly what the question is asking: Subtract (5x^2 - 4) - (2x^2 + 3x - 9).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Distribute the subtraction sign.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 3x^2 - 3x + 5

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