Binomial and special products
Distribution gives every product. Special identities collect those products when the brackets match a precise pattern
- Every product Each term multiplies every term in the other bracket. Collect only like powers \(\displaystyle \begin{aligned}&(a+b)(c+d)\\&=ac+ad+bc+bd\end{aligned}\)
- Square of a sum Two equal cross-products give the middle term \(\displaystyle (a+b)^2=a^2+2ab+b^2\)
- Square of a difference The middle term is −2ab; the last term is +b². Carry any signs already in a and b through those products \(\displaystyle (a-b)^2=a^2-2ab+b^2\)
- Sum times difference The cross-products cancel. This is not a squared difference \(\displaystyle (a+b)(a-b)=a^2-b^2\)
Worked example
Expand and simplify the squared difference
- \(\displaystyle (3x-4)(3x-4)\)A square repeats the entire bracket
- \(\displaystyle 9x^2-12x-12x+16\)Form all four products, including both cross-products
- \(\displaystyle -12x-12x=-24x\)Collect the like terms
- \(\displaystyle 9x^2-24x+16\)The constant is positive because both factors are negative
- \(\displaystyle x=2:\quad(6-4)^2=4\)Check the original at x = 2
- \(\displaystyle 36-48+16=4\)The expansion agrees at that value. Distribution establishes the identity
Common mistakes
Difficulty levels
Questions about this topic
Is (a − b)² the same as a² − b²?+
No. The squared difference has the middle term −2ab and ends with +b². The sum-times-difference product gives a² − b²
Can coefficients be negative?+
Yes. Keep each sign through every multiplication and when collecting like powers
Why does the picture use positive lengths?+
A physical area model requires positive lengths. The distributive identity remains valid for negative and zero algebraic inputs
Can substitution prove an identity?+
One matching input is a check, not a proof. Distribution proves the identity for all inputs; one disagreement disproves a proposed identity
Why is the cut-out restricted to x ≥ 5?+
Its side x must fit within 2x − 5. Thus x ≤ 2x − 5, or x ≥ 5. A positive rectangle alone is not enough
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 10: Quadratic and non-linear graphs