Notice the structure: the powers of \(a\) step down (\(3, 2, 1, 0\)), the powers of \(b\) step up (\(0, 1, 2, 3\)), and the coefficients are \(1, 3, 3, 1\).
The square of a trinomial
Multiplying \((a + b + c)(a + b + c)\) gives nine products, which combine into six terms.
There are three pairs (first-second, first-third, second-third), so there are three doubled products. Each one takes whatever sign the multiplication gives it.
Practice
Quick check
What is the third term of \((x - 4)^3 = x^3 - 12x^2 + \ldots\)?
It's three times the first term times the last term squared: \(3 \cdot x \cdot (-4)^2 = 48x\). The square makes it positive.
Quick check
How many terms are in the expanded form of \((a + b + c)^2\), before any like terms are combined?
Three squares plus three doubled products. The nine raw products collapse to six because pairs like \(ab\) and \(ba\) merge.
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