The triangle diagram for the square of a trinomial

Observation — Triangle diagram for $(A+B+C)^2$

To recognise whether a 6-term polynomial is a square of a trinomial, a triangle diagram is used: the squares of the three terms at the vertices, the double products on the sides.

At the vertices the squares of the terms, on the sides their double products.

Example. $1+4t^2+4z^2-4t+4z-8tz$. I suspect a square of a trinomial.

• Vertices: $1 = (1)^2$, $\;4t^2 = (2t)^2$, $\;4z^2 = (2z)^2$.
• Sides: $2\cdot 1\cdot 2t = 4t$? The term is $-4t$, so the sign is $-$.
• $2\cdot 1\cdot 2z = 4z$. ✓
• $2\cdot 2t\cdot 2z = 8tz$? The term is $-8tz$, so the sign is $-$.

Therefore $1+4t^2+4z^2-4t+4z-8tz = \boxed{(1-2t+2z)^2}$.

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Topics: Algebraic calculus
Concepts: Special products · Square of a trinomial
Skills: Using formulas