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
- Add and subtract polynomials
- Multiply polynomial expressions
- Track degree
Worked example
Problem. Simplify (3x^2 + 4x - 1) + (2x^2 - 9x + 5).
- Combine x^2 terms.
- Combine x terms.
- Combine constants.
Answer: 5x^2 - 5x + 4
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Simplify (2x + 7) + (5x - 3).
Show solution
- Combine like terms.
Answer: 7x + 4
2. Simplify (6x^2 - x) - (2x^2 + 5x).
Show solution
- Subtract each term.
Answer: 4x^2 - 6x
3. The degree of 9x^5 - 3x^2 + 1 is...
Choices: 5 · 9 · 2 · 1
Show solution
- Degree is the highest exponent.
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
- Combine like terms.
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
- Distribute the subtraction sign.
Answer: 3x^2 - 3x + 5
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