Math101 · Independent Ontario learning library
Math101Special Products
Special products are recurring polynomial multiplication patterns: $(a+b)^2=a^2+2ab+b^2$, $(a-b)^2=a^2-2ab+b^2$, and $(a+b)(a-b)=a^2-b^2$.
These identities speed exact calculation, expansion, factoring, completing squares, and manipulation of conjugates while revealing geometric structure.
Intuition and core definition
Special products are recurring polynomial multiplication patterns: $(a+b)^2=a^2+2ab+b^2$, $(a-b)^2=a^2-2ab+b^2$, and $(a+b)(a-b)=a^2-b^2$. They are consequences of distribution, not separate exceptions.
Notation, language, and conditions
Conjugates differ only in the sign between terms. The square pattern has three terms because the two cross-products combine. Variables, coefficients, and larger expressions may occupy $a$ and $b$ as long as grouping is preserved.
Why this idea matters
Special-product identities compress predictable multiplication patterns while remaining consequences of ordinary distribution.
A dependable method
- Identify whether factors are identical binomials or conjugates.
- Assign complete expressions to $a$ and $b$, including coefficients.
- Apply the matching identity with the correct middle sign.
- Simplify powers and coefficient products.
- Expand by distribution as a check.
