Algebra
Polynomials
Special products
20 practice questions
2 video lessons
Theory + worked examples
Theory
Special products are patterns worth memorizing:
- Square of a sum: \((a+b)^2=a^2+2ab+b^2\).
- Square of a difference: \((a-b)^2=a^2-2ab+b^2\).
- Difference of squares: \((a+b)(a-b)=a^2-b^2\).
\((a+b)^2\neq a^2+b^2\) — the middle term \(2ab\) matters.
The special product patterns.
Worked examples.
The patterns:
\[(a\pm b)^2=a^2\pm2ab+b^2,\qquad (a+b)(a-b)=a^2-b^2\]
The difference of squares has no middle term.
How to use special products
- Recognize the pattern.
- Identify \(a\) and \(b\).
- Apply the formula directly.
- Simplify coefficients.
Example 1 — Square of a sum
Expand \((x+3)^2\).
Solution
Use \((a+b)^2=a^2+2ab+b^2\).
| \((x+3)^2\) | \(=\) | \(x^2+6x+9\) |
Example 2 — Square of a difference
Expand \((x-3)^2\).
Solution
Use \((a-b)^2=a^2-2ab+b^2\).
| \((x-3)^2\) | \(=\) | \(x^2-6x+9\) |
Example 3 — Difference of squares
Expand \((x+3)(x-3)\).
Solution
The middle terms cancel.
| \((x+3)(x-3)\) | \(=\) | \(x^2-9\) |
Example 4 — With a coefficient
Expand \((2x+1)^2\).
Solution
Apply the square pattern.
| \((2x+1)^2\) | \(=\) | \(4x^2+4x+1\) |
Common pitfalls
\((a+b)^2=a^2+2ab+b^2\), not \(a^2+b^2\).
Square the coefficient too: \((2x)^2=4x^2\).
Difference of squares loses the middle term.
Frequently asked questions
What is \((a+b)^2\)?
\(a^2+2ab+b^2\).
What is \((a+b)(a-b)\)?
\(a^2-b^2\), the difference of squares.
Is \((x+2)^2=x^2+4\)?
No — it is \(x^2+4x+4\).
Why memorize special products?
They speed up expanding and factoring.
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