Tartaglia's triangle

Observation — Tartaglia's triangle (Pascal)

The coefficients of the expansion of $(\textcolor{#1f77b4}{\bigcirc}+\textcolor{#d62728}{\square})^n$ can be read off from Tartaglia’s triangle: each number is the sum of the two above it.

Each number is the sum of the two above it; row $n$ gives the coefficients of $(\bigcirc+\square)^n$.

Example. To expand $(2x+3y)^4$, we read row $n=4$, that is $1,4,6,4,1$: $(2x+3y)^4 = 1\cdot(2x)^4 + 4\cdot(2x)^3(3y) + 6\cdot(2x)^2(3y)^2 + 4\cdot(2x)(3y)^3 + 1\cdot(3y)^4$ $= 16x^4 + 96x^3y + 216x^2y^2 + 216xy^3 + 81y^4$

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Topics: Algebraic calculus
Concepts: Cube of a binomial · Special products · Tartaglia’s triangle
Skills: Using formulas
People: Blaise Pascal · Niccolò Tartaglia