Special Products of Polynomials

Let's now see how special products of polynomials apply to geometry problems and to mental arithmetic. Look at the following example.

Example: Find the area of the square.

Solution: \(\begin{align*}The \ area \ of \ the \ square = side \times side\end{align*}\)

\(\begin{align*}\text{Area} & = (a+b)(a+b)\\ & = a^2+2ab+b^2\end{align*}\)

Notice that this gives a visual explanation of the square of binomials product.

\(\begin{align*}Area \ of \ big \ square: (a+b)^2 = Area \ of \ blue \ square = a^2+2 \ (area \ of \ yellow) = 2ab + area \ of \ red \ square = b^2\end{align*}\)

The next example shows how to use the special products in doing fast mental calculations.

Example 3: Find the products of the following numbers without using a calculator.

(a) \(\begin{align*}43 \times 57\end{align*}\)

(b) \(\begin{align*}45^2\end{align*}\)

Solution: The key to these mental "tricks" is to rewrite each number as a sum or difference of numbers you know how to square easily.

(a) Rewrite \(\begin{align*}43=(50-7)\end{align*}\) and \(\begin{align*}57=(50+7)\end{align*}\).

Then \(\begin{align*}43 \times 57 = (50-7)(50+7) = (50)^2-(7)^2=2500-49=2,451\end{align*}\).

(b) \(\begin{align*}45^2 = (40+5)^2 = (40)^2+2(40)(5) +(5)^2 = 1600+400+25=2,025\end{align*}\)