46.3 Square of a Binomial Difference

Square of a Binomial Difference is the special product formed when a two-term expression joined by subtraction is multiplied by itself, producing a three-term result known as a perfect square trinomial with a negative middle term. 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, minus twice the product of the two terms, plus the square of the second term, assembled in that fixed order.

This pattern closely parallels the square of a binomial sum, differing only in the sign of the middle term, and it is likewise a direct consequence of applying the distributive property to identical binomial factors.

Recognizing the Pattern

Repeated Difference Binomial Recognition

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

Building the Expansion Term by Term

First Difference 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, exactly as in the binomial sum case.

Negative Double Product Middle Term

The middle term arises from the two cross multiplications in the distributive expansion: a multiplied by −b gives −ab, and −b multiplied by a gives another −ab. Because both cross terms are identical and negative, combining them produces a coefficient of negative two rather than positive two.

Subtracted Term Square

The final multiplication in the expansion is −b multiplied by −b. Because a negative multiplied by a negative produces a positive result, this term becomes +b² rather than a negative quantity, even though the original binomial term was subtracted.

Assembling the Complete Pattern

Square-of-Difference Expansion Assembly

Combining the first term square, the negative middle term, and the final positive square term in the order they were produced gives the complete expansion of the square of a binomial difference.

Positive Final Square Term

A frequent point of confusion is expecting the final term to carry a negative sign because the original binomial subtracted b; however, since that term is squared, and the square of any real quantity, positive or negative, is never negative, the final term of the trinomial is always positive.

Square-of-Difference Standard Form

The standard form of this expansion places the squared terms at the two ends, both positive, with the doubled product in the middle carrying a negative sign, giving a trinomial shape that differs from the binomial-sum pattern only in that one sign.

Comparing to the Binomial Sum Pattern

Shared and Differing Structure

Both the square of a sum and the square of a difference produce a first term equal to a² and a last term equal to b²; the two patterns differ solely in the sign of the middle term, positive for a sum and negative for a difference, since that sign is inherited directly from the sign joining the original binomial's two terms.

Worked Example

Numerical Application

Applying the pattern to (x − 4)² identifies a as x and b as 4. The first term square is x², the middle term is −2 · x · 4 = −8x, and the final term is 4² = 16, all positive despite the original subtraction.