Square of a Trinomial- Formula and Examples

What is the Square of a Trinomial?

A trinomial is simply an algebraic expression with three terms. When you square it, you multiply it by itself. That's all that's happening here.

The general form is:

(a + b + c)²

This equals (a + b + c)(a + b + c). You're doing the same multiplication you learned in basic algebra, just with more terms.

The Formula

When you expand (a + b + c)², you get:

(a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc

Three things get squared individually (a², b², c²). Then you get six cross-products, but since ab = ba and so on, they collapse into three terms with coefficients of 2.

Step-by-Step Expansion

Here's how to work it out manually:

• Step 1: Write it as multiplication: (a + b + c)(a + b + c)
• Step 2: Multiply each term in the first bracket by each term in the second bracket
• Step 3: Collect like terms
• Step 4: Simplify

Working through it:

a × a = a²
a × b = ab
a × c = ac
b × a = ab
b × b = b²
b × c = bc
c × a = ac
c × b = bc
c × c = c²

Combine: a² + b² + c² + 2ab + 2ac + 2bc

Examples

Example 1: (x + y + z)²

Using the formula directly:

= x² + y² + z² + 2xy + 2xz + 2yz

That's your answer. No extra steps needed once you know the pattern.

Example 2: (2x + 3y + 4)²

Here a = 2x, b = 3y, c = 4

= (2x)² + (3y)² + (4)² + 2(2x)(3y) + 2(2x)(4) + 2(3y)(4)

= 4x² + 9y² + 16 + 12xy + 16x + 24y

Check: 2(2x)(3y) = 12xy ✓

Example 3: (x - 2y + 3z)²

Negative signs work the same way. Just carry them through.

a = x, b = -2y, c = 3z

= x² + (-2y)² + (3z)² + 2(x)(-2y) + 2(x)(3z) + 2(-2y)(3z)

= x² + 4y² + 9z² - 4xy + 6xz - 12yz

Notice: (-2y)² = 4y². Squaring a negative gives a positive.

Common Mistakes to Avoid

• Forgetting to double the cross terms. The coefficient is always 2, not 1. Some students write ab + ac + bc and miss the factor of 2.
• Squaring the negative wrong. (-b)² = b². The negative disappears when you square.
• Skipping terms. Nine products come from the FOIL-style expansion. Make sure you account for all of them.
• Not combining like terms. ac and ca are the same thing. Add them together.

Quick Reference Table

Practice: Expand (x + 2 + 3y)²

Try it yourself before looking.

Solution: a = x, b = 2, c = 3y

= x² + 4 + 9y² + 4x + 6xy + 12y

The constant term 2 squared gives 4. The cross term 2(x)(3y) = 6xy. Everything else follows the pattern.

When You'll Actually Use This

Factoring trinomial squares works in reverse. If you see x² + y² + z² + 2xy + 2xz + 2yz, you know it factors back to (x + y + z)². Recognizing this pattern saves time in algebra and calculus problems.

That's it. Memorize the pattern, apply it to examples, and move on.