4.5.2 The FOIL Method
Using the FOIL method: "First, Inner, Outer, Last", we can multiply out any two binomials. The process is developed using the distributive property of numbers.
The geometric approach of multiplying out binomials is great if we can use a square or rectangle to model the process but not so great if the binomial expressions are less than or equal to 0 — you can't have a negative side of a rectangle. However, if we think about things from an algebraic point of view, we can use the power of the distributive property of numbers to develop a method that will always work. Recall that the distributive property for numbers tells that:
a(b+c)=ab+ac
To multiply out (2x+3)(x+2), we can let a=(2x+3) and b=x while we can let c=2. We then have that
\begin{align*}a(b+c)&=(2x+3)(x+2)\\&=(2x+3)x+(2x+3)2\\&=2x^2+3x+4x+6 \\&=2x^2+7x+6\end{align*}
If you multiply out a bunch of binomials, you'll begin to notice a pattern; the final sum is made up of the product of the First terms of the two binomials plus the product of the two Outer terms, plus the product of the two Inner terms, plus the product of the two Last terms. In other words, by using the FOIL method,
(2x+3)(x+2)=2x^2+4x+3x+6=2x^2+7x+6.
This way of thinking about the multiplication of binomials is oftentimes much quicker than thinking through the distribution. And, though it's easy to apply, it's also easy to forget why it works. Try to remind yourself while practicing the FOIL method that it was developed using the distributive property of numbers.
Explore!
| First | Outer | Inner | Last | Combined Expression | |
|---|---|---|---|---|---|
| (2x-3)(x-5) | 2x^2 | 2x^2-13x+15 |
| First | Outer | Inner | Last | Combined Expression | |
|---|---|---|---|---|---|
| (x-2)(x+5) | x^2 | x^2+3x-10 |
Now that we know how to multiply out binomials, we can develop a sense for the reverse process: factoring complicated quadratics. Then, we'll be able to see if have enough tools to answer our original question, "When will the egg hit the ground?" whose mathematical equation is given by: -4.9x^{2}+2.45x+1.7=0
We'll study more advanced factoring in the next section.
