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
- Multiply binomials fluently
- Use square and difference patterns
- Connect products to factoring
Worked example
Problem. Expand (x - 4)(x + 4).
- This is a difference of squares pattern.
- Use (a - b)(a + b) = a^2 - b^2.
- x^2 - 4^2 = x^2 - 16.
Answer: x^2 - 16
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. Expand (x + 3)(x + 5).
Choices: x^2 + 8x + 15 · x^2 + 15 · x^2 + 2x + 15 · 2x + 8
Show solution
- The middle terms are 5x and 3x.
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
- Difference of squares.
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
- The middle term is 2 x x x 4.
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
- The middle terms are 14x and -3x.
Answer: 2x^2 + 11x - 21
5. Simplify (2x + 3) + (x - 5).
Show solution
- Combine the x terms: 2x + x = 3x.
- Combine constants: 3 - 5 = -2.
- The simplified expression is 3x - 2.
Answer: 3x - 2
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