Multiplying Special Cases

Square of a Binomial

The Square of a binomial looks like

    \[(a+b)^2\]

    \[(a+b)^2 \neq (a^2+b^2)\]

Think Pythagorean Theorem for the reason.

To obtain the correct value we need to FOIL. The meaning of (x+5)^2 is (x+5)(x+5)

    \[(x+5)^2= x^2+10x+25\]

General Rule: (a+b)^2= \mathbf{a^2+2ab+b^2}

Example: (2q-9)^2=4q^2-36q+81

Use this pattern to simply convert the square of a binomial without actually FOILing.

 

 

Difference of Squares

    \[a^2-b^2\]

This is called the \textbf{difference of squares} because the a is squared, the b is also squared, and they are subtracted (difference).

This happens when you multiply 2 special binomials.

    \[(a+b)(a-b)= \mathbf{a^2-b^2}\]

Example: Foil the following (x+2)(x-2)

    \[(x^2-2x+2x-4)\]

Notice the middle terms (-2x+2x) cancel out. The first term x^2 and last term 4 are both perfect squares.  They have subtraction in between them, hence the “difference” in difference of squares.

Thus, our answer is

    \[\mathbf{x^2-4}\]