46.2 Square of a Binomial Sum

The square of a binomial sum expands to a trinomial using the distributive property, revealing patterns essential in algebraic simplification.

Square of a Binomial Sum is the special product formed when a two-term expression joined by addition is multiplied by itself, producing a three-term result known as a perfect square trinomial. It describes the specific pattern that arises from (a + b) multiplied by (a + b), where the outcome always consists of the square of the first term, twice the product of the two terms, and the square of the second term, assembled in that fixed order.

This pattern is a direct consequence of the distributive property applied to identical binomial factors, and it provides a shortcut that avoids re-deriving the full expansion from scratch each time an addition binomial is squared.

Recognizing the Pattern

Repeated Sum Binomial Recognition

Before applying the shortcut, the expression must be confirmed to be a binomial addition multiplied by itself, meaning both factors are identical and the terms within the binomial are joined by a plus sign. An expression such as (a + b)(a + b) or its equivalent notation (a + b)² both qualify, while an expression with two different binomials or a subtraction does not follow this exact pattern.

(a + b)^{2} = (a + b) (a + b)

Building the Expansion Term by Term

First Binomial Term Square

The first term of the resulting trinomial is the square of the first term of the binomial, obtained by multiplying that term by itself during the distributive expansion.

a \cdot a = a^{2}

Positive Double Product Middle Term

The middle term arises from the two cross multiplications in the distributive expansion, a multiplied by b and b multiplied by a, both of which are identical and both positive because the original binomial contains only addition. Combining these two identical terms produces a coefficient of two.

a \cdot b + b \cdot a = 2 a b

Second Binomial Term Square

The final term of the trinomial is the square of the second term of the binomial, obtained the same way as the first term square, by multiplying that term by itself.

b \cdot b = b^{2}

Assembling the Complete Pattern

Square-of-Sum Expansion Assembly

Combining the first term square, the middle term, and the second term square in the order they were produced gives the complete expansion of the square of a binomial sum.

(a + b)^{2} = a^{2} + 2 a b + b^{2}

Like-Term Reduction in the Middle Position

Without recognizing the pattern, the full expansion of (a + b)(a + b) initially produces four partial products: a², ab, ba, and b². Since ab and ba represent the same quantity, like-term reduction combines them into the single middle term 2ab, which is why the final result has three terms rather than four.

Square-of-Sum Standard Form

The standard form of this expansion always places the squared terms at the two ends and the doubled product in the middle, arranged from the term derived from the first binomial element to the term derived from the second, giving a consistent, recognizable trinomial shape.

Worked Example

Numerical Application

Applying the pattern to (x + 5)² identifies a as x and b as 5. The first term square is x², the middle term is 2 · x · 5 = 10x, and the second term square is 5² = 25.

(x + 5)^{2} = x^{2} + 10 x + 25

Visual Model of the Pattern

The square with side length a + b divides into a square of area a², a square of area b², and two identical rectangles each of area ab, whose combined area of 2ab corresponds exactly to the middle term of the algebraic expansion.