Algebra 2 Polynomial Operations Worksheet with Answers

Polynomial operations are among the most extensively tested topics in Algebra 2 — and they form the foundation for higher-level work in rational expressions, function analysis, and calculus. Students who struggle with polynomial multiplication or factoring will hit a wall in these later topics, which is why systematic practice at this stage is so valuable.

This worksheet covers the five core polynomial operation types: adding, subtracting, multiplying, dividing (polynomial long division), and factoring. Each section includes examples followed by practice problems, with a complete answer key at the end.

Section 1: Adding and Subtracting Polynomials

To add or subtract polynomials, combine like terms — terms with the same variable and exponent. When subtracting, distribute the negative sign before combining.

Example: (3x² + 2x − 5) + (x² − 4x + 3)

Solution: Combine like terms: (3x² + x²) + (2x − 4x) + (−5 + 3) = 4x² − 2x − 2.

Example: (5x² − 3x + 7) − (2x² + x − 4)

Solution: Distribute minus: 5x² − 3x + 7 − 2x² − x + 4 = 3x² − 4x + 11.

Practice:

1. (4x³ − 2x + 1) + (x³ + 5x − 3)
2. (7x² + 3x − 9) − (4x² − x + 2)
3. (2x³ + x² − 5) + (−x³ + 3x² + 2x)

Section 2: Multiplying Polynomials

Multiply polynomials by applying the distributive property to every term in the first polynomial with every term in the second. For binomials, FOIL (First, Outer, Inner, Last) is a useful memory aid — but it only works for two binomials. For larger polynomials, use the full distributive property.

Example: (x + 3)(x − 5)

Solution (FOIL): x·x + x·(−5) + 3·x + 3·(−5) = x² − 5x + 3x − 15 = x² − 2x − 15.

Example: (2x − 1)(3x² + x − 4)

Solution: Distribute 2x: 6x³ + 2x² − 8x. Distribute −1: −3x² − x + 4. Combine: 6x³ + 2x² − 3x² − 8x − x + 4 = 6x³ − x² − 9x + 4.

Example: (x + 4)²

Solution: Use (a + b)² = a² + 2ab + b²: x² + 8x + 16.

Practice:

4. (x − 2)(x + 7)
5. (3x + 1)(2x − 5)
6. (x + 3)²
7. (x − 4)(x² + 2x − 3)

Section 3: Polynomial Long Division

Dividing a polynomial by a binomial uses the same process as numeric long division. You divide the leading term, multiply, subtract, bring down the next term, and repeat until no remainder.

Example: Divide (x² + 5x + 6) ÷ (x + 2)

Solution:

         x + 3
       ---------
x + 2 | x² + 5x + 6
         x² + 2x
         ---------
              3x + 6
              3x + 6
              ------
                 0

Answer: x + 3 (no remainder).

Example: Divide (2x² − 3x − 5) ÷ (x − 2)

Solution:

           2x + 1
         ---------
x − 2 | 2x² − 3x − 5
         2x² − 4x
         ---------
               x − 5
               x − 2
               -----
                 −3

Answer: 2x + 1 remainder −3, written as 2x + 1 − 3/(x − 2).

Practice:

8. (x² + 7x + 12) ÷ (x + 3)
9. (3x² − x − 10) ÷ (x − 2)

Section 4: Factoring Polynomials

Factoring is the reverse of multiplication. Always check for a GCF first, then look for the appropriate factoring pattern.

GCF Factoring Example: 6x³ − 9x² + 3x

Solution: GCF = 3x. Factor: 3x(2x² − 3x + 1).

Trinomial (a = 1) Example: x² + 7x + 12

Solution: Find two numbers that multiply to 12 and add to 7: 3 and 4. Answer: (x + 3)(x + 4).

Trinomial (a ≠ 1) Example: 2x² + 7x + 3

Solution: AC method: a × c = 6. Find two numbers multiplying to 6 and adding to 7: 1 and 6. Rewrite: 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1).

Difference of Squares Example: x² − 25

Solution: a² − b² = (a + b)(a − b). Answer: (x + 5)(x − 5).

Perfect Square Trinomial Example: x² − 6x + 9

Solution: Recognizes as (x − 3)² since (−3)² = 9 and 2·(−3) = −6.

Practice:

10. x² − x − 12
11. 3x² + 11x + 6
12. 4x² − 49
13. x² + 10x + 25
14. 5x³ − 15x² + 10x

Answer Key

Section 1:

1. 5x³ + 3x − 2
2. 3x² + 4x − 11
3. x³ + 4x² + 2x − 5

Section 2:

4. x² + 5x − 14
5. 6x² − 13x − 5
6. x² + 6x + 9
7. x³ − 2x² − 11x + 12

Section 3:

8. x + 4 (no remainder)
9. 3x + 5 (no remainder)

Section 4:

10. (x − 4)(x + 3)
11. (3x + 2)(x + 3)
12. (2x + 7)(2x − 7)
13. (x + 5)²
14. 5x(x² − 3x + 2) = 5x(x − 1)(x − 2)

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