Example 6.5.2

To “square a binomial” is to take a binomial and multiply it by itself. We know that exponent notation means that \(4^2=4\cdot 4\text{.}\) Applying this to a binomial, we'll see that \((x+4)^2=(x+4)(x+4)\text{.}\) To expand this expression, we'll simply distribute \((x+4)\) across \((x+4)\text{:}\)

\begin{align*} \left( x+4 \right)^2 \amp= \left( x+4 \right)\left( x+4 \right) \\ \amp= x^2 + 4x + 4x + 16 \\ \amp= x^2 + 8x + 16 \end{align*}

Similarly, to expand \((y-7)^2\text{,}\) we'll have:

\begin{align*} \left( y-7 \right)^2 \amp= \left( y-7 \right)\left( y-7 \right)\\ \amp= y^2 -7y -7y + 49 \\ \amp= y^2 -14y + 49 \end{align*}

These two examples might look like any other example of multiplying binomials, but looking closely we can see that something very specific (or special) happened. Focusing on the original expression and the simplified one, we can see that a specific pattern occurred in each:

\begin{align*} \left( x+4 \right)^2 \amp= x^2 + \highlight{4}x + \highlight{4}x + \highlight{4\cdot 4}\\ \left( x+\highlight{4} \right)^2 \amp= x^2 +2(\highlight{4}x) + \highlight{4}^2\\ \end{align*}


\begin{align*} \left( y-7 \right)^2 \amp= y^2 -\highlight{7}y - \highlight{7}y + \highlight{7\cdot 7} \\ \left( y-\highlight{7} \right)^2 \amp= y^2 -2(\highlight{7}y) + \highlight{7}^2 \end{align*}

Notice that the two middle terms are not only the same, they are also exactly the product of the two terms in the binomial. Furthermore, the last term is the square of the second term in each original binomial.