Squaring a Trinomial: Formula and Examples

TL;DR

Squaring a trinomial applies the identity (a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ca, three squares plus twice each pair-product. This article derives the formula, works through examples with numbers and variables, and shows why the cross terms are the part everyone forgets.

What Is Squaring a Trinomial?

A trinomial is a polynomial with exactly three terms, for example x+y+z or x² + 2x + 3. Squaring it means raising the whole expression to the power 2, so (a+b+c)² is shorthand for (a+b+c)(a+b+c).

The result is governed by one of the standard algebraic identities: (a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ca.

Two things are worth naming right away. The squared terms a², b², c² come from each term meeting itself. The cross terms 2ab, 2bc, 2ca come from each pair of different terms, and the factor of 2 appears because each pair shows up twice in the full multiplication.

How Do You Derive the Squaring a Trinomial Formula?

Treat (a+b+c)(a+b+c)(a+b+c) as (a+(b+c)) and use the binomial square: (a+(b+c))² = a² + 2a(b+c) + (b+c)².

Expand the last piece with (b+c)² = b² + 2bc + c²: a² + 2ab + 2ac + b² + 2bc + c².

Reorder into squares first, then pairs: a² + b² + c² + 2ab + 2bc + 2ca.

That is the identity. Every squared term appears once; every cross term appears with a 2. This is the same multiplying-polynomials skill applied to a three-term factor.

Examples of Squaring a Trinomial

Example 1

Expand (x+y+z)².
Square each term: x², y², z².
Twice each pair: 2xy, 2yz, 2zx.

(x+y+z)² = x² + y² + z² + 2xy + 2yz + 2zx.
Final answer: as above.

Example 2

A student expands (2+3+4)² as 2² + 3² + 4² = 29.
The tempting move is to square each term and add, it feels complete.

That gives 4 + 9 + 16 = 29.
Check it against the direct value: 2 + 3 + 4 = 9, and 9² = 81, not 29.

The gap is the cross terms. Add them: 2(2⋅3) + 2(3⋅4) + 2(4⋅2) = 12 + 24 + 16 = 52.

Now 29 + 52 = 81.
Final answer: 81. Squaring the terms and stopping drops the six cross-terms, and the answer is off by more than half.

Example 3

Expand (x + 2y + 3)².
Set a = x, b = 2y, c = 3.

Squares: x², (2y)² = 4y², 3² = 9. Pairs: 2(x)(2y) = 4xy, 2(2y)(3) = 12y, 2(3)(x) = 6x.

(x + 2y + 3)² = x² + 4y² + 9 + 4xy + 12y + 6x.
Final answer: as above.

Example 4

Expand (a+b−c)².
A subtracted term is a negative term, so set c→−c.

Squares: a², b², (−c)² = c². Pairs: 2ab, 2(b)(−c) = −2bc, 2(−c)(a) = −2ca.

(a+b−c)² = a² + b² + c² + 2ab − 2bc − 2ca.
Final answer: the cross terms involving c flip sign; the squares stay positive.

Example 5

Expand (x² + 2x + 3)².
Set a = x², b = 2x, c = 3.

Squares: x⁴, 4x², 9. Pairs: 2(x²)(2x) = 4x³, 2(2x)(3) = 12x, 2(3)(x²) = 6x². Combine the x² terms: 4x² + 6x² = 10x².

(x² + 2x + 3)² = x⁴ + 4x³ + 10x² + 12x + 9.
Final answer: as above.

Example 6

A square field has side (p+q+5) metres. Write its area.
Area of a square is side squared.

(p+q+5)² = p² + q² + 25 + 2pq + 10q + 10p.
Final answer: p² + q² + 25 + 2pq + 10p + 10q square metres.

Why the Cross Terms Are the Whole Point

The formula matters because the cross terms carry real quantity, they are not decoration.

• In geometry, they are area. Split a square of side (a+b+c) into a grid, and the cross terms are the rectangular strips between the corner squares. Leave them out and you have lost pieces of the square.
• In algebra, they connect to the perfect square trinomial. The binomial case is the same idea with one cross term; the trinomial has three.
• In error analysis, dropped cross terms compound. When engineers linearise a formula and discard the "small" cross terms too early, the leftover error grows, as seen in the Tacoma Narrows Bridge collapse of 1940.

Where the Expansion Goes Wrong

Every common error traces back to the cross terms.

Mistake 1: Distributing the square over the sum

Where it slips in: the very first step.

Don't do this: write (a+b+c)²=a²+b²+c².
The correct way: a square never distributes over addition.

Mistake 2: Missing a pair

Where it slips in: listing the cross terms from memory.

Don't do this: write only 2ab + 2bc and forget 2ca.

Mistake 3: Mishandling a negative term

Where it slips in: expanding (a−b+c)².

Don't do this: treat −b as b and copy the all-positive formula.
The correct way: carry the sign into the term.

Conclusion

• Squaring a trinomial uses the identity (a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ca.
• The three squares are straightforward; the three doubled cross-terms are frequently forgotten.
• A negative term keeps its square positive but flips the sign of its cross-products.