CMClearMathAcademy

Binomial Products and Special Patterns

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

Polynomial arithmetic gets faster when you recognize common binomial patterns. Difference of squares and perfect-square trinomials appear often in factoring, equations, and graph analysis.

What you'll learn

Why it matters: Expanded box dimensions, two-period compound growth, and engineering tolerance ranges all multiply binomials. FOIL — First, Outer, Inner, Last — is the systematic four-multiplication map that guarantees you get every term.

Worked example

Problem. Expand (x - 4)(x + 4).

  1. This is a difference of squares pattern.
  2. Use (a - b)(a + b) = a^2 - b^2.
  3. x^2 - 4^2 = x^2 - 16.

Answer: x^2 - 16

Practice problems

1. Expand (x + 3)(x + 5).

Choices: x^2 + 8x + 15 · x^2 + 15 · x^2 + 2x + 15 · 2x + 8

Show solution
  1. Warm-up: First identify exactly what the question is asking: Expand (x + 3)(x + 5).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The middle terms are 5x and 3x.
  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 + 8x + 15

2. Expand (x - 6)(x + 6).

Choices: x^2 - 36 · x^2 + 36 · x^2 - 12x + 36 · x^2 + 12x + 36

Show solution
  1. Warm-up: First identify exactly what the question is asking: Expand (x - 6)(x + 6).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Difference of squares.
  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 - 36

3. Expand (x + 4)^2.

Choices: x^2 + 8x + 16 · x^2 + 16 · x^2 + 4x + 16 · x^2 - 8x + 16

Show solution
  1. Core Practice: First identify exactly what the question is asking: Expand (x + 4)^2.
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The middle term is 2 x x x 4.
  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 + 8x + 16

4. Expand (2x - 3)(x + 7).

Choices: 2x^2 + 11x - 21 · 2x^2 + 14x - 21 · 2x^2 - 17x - 21 · 3x^2 + 4x - 21

Show solution
  1. Challenge: First identify exactly what the question is asking: Expand (2x - 3)(x + 7).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The middle terms are 14x and -3x.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 2x^2 + 11x - 21

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

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

Answer: 3x - 2

Practice this interactively with instant feedback and an AI tutor.

Practice Binomial Products and Special Patterns Take the free placement check

More Algebra II lessons