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Polynomial Operations

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

Polynomials are expressions made of terms with whole-number exponents. Add and subtract by combining like terms. Multiply using distribution and then combine like terms.

What you'll learn

Why it matters: Variable-side area sums, physics displacement totals, and stacked-fee budget rules all combine polynomial terms. Combining like terms collapses the expression into a single readable form so the next step is easier.

Worked example

Problem. Simplify (3x^2 + 5x - 2) + (x^2 - 7x + 9).

  1. Combine x^2 terms: 3x^2 + x^2 = 4x^2.
  2. Combine x terms: 5x - 7x = -2x.
  3. Combine constants: -2 + 9 = 7.

Answer: 4x^2 - 2x + 7

Practice problems

1. Simplify (2x + 5) + (3x - 8).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Simplify (2x + 5) + (3x - 8).
  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 5x - 3 and make sense in the original problem.

Answer: 5x - 3

2. Simplify (4x^2 - x) + (2x^2 + 7x).

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

Answer: 6x^2 + 6x

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: Simplify (5x^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. Distribute the subtraction first.
  4. Check the result by substituting or estimating: the response should match 3x^2 + 4x - 5 and make sense in the original problem.

Answer: 3x^2 + 4x - 5

4. Multiply (x + 4)(x + 5).

Choices: x^2 + 9x + 20 · x^2 + 20 · x^2 + x + 20 · 2x + 9

Show solution
  1. Core Practice: First identify exactly what the question is asking: Multiply (x + 4)(x + 5).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Use distribution.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x^2 + 9x + 20

5. The degree of 7x^4 - 3x^2 + 8 is...

Choices: 4 · 7 · 2 · 8

Show solution
  1. Challenge: First identify exactly what the question is asking: The degree of 7x^4 - 3x^2 + 8 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: 4

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