Chapter A.3, Problem 142AYU

To determine

To find: Do you prefer to memorize the rule for the square of a binomial ${\left(x+a\right)}^2$ or to use FOIL to obtain the product? Write a brief position paper defending your choice.

Answer to Problem 142AYU

There is also a special method, useful ONLY for a two-term polynomial times another two-term polynomial. The method is called "FOIL". The letters F-O-I-L come from the words "first", "outer", "inner", "last", and are a memory device for helping you remember how to multiply horizontally, without having to write out the distribution like I did, and without dropping any terms. Here is what FOIL stands for:

That is, FOIL tells you to multiply the first terms in each of the parentheses, then multiply the two terms that are on the "outside" (furthest from each other), then the two terms that are on the "inside" (closest to each other), and then the last terms in each of the parentheses. In other words, using the previous example:

Use FOIL to simplify $\left(x +3\right)\left(x +2\right)$

Okay; the instructions specify the method I have to use, so here goes:

"first": $\left(x\right)\left(x\right)= x^2$

"outer": $\left(x\right)\left(2\right)=2x$

"inner": $\left(3\right)\left(x\right)=3x$

"last": $\left(3\right)\left(2\right)=6$

Adding the results of the four multiplications together, and combining the two "like" terms in the middle, I get:

$x^2 +2x +3x +6$

$x^2 +5x +6$

I prefer the FOIL to obtain product instead of the rule for the square of a binomial ${\left(x+a\right)}^2$. Since FOIL is used for all kind of product and it is very easy to use.

Explanation of Solution

There is also a special method, useful ONLY for a two-term polynomial times another two-term polynomial. The method is called "FOIL". The letters F-O-I-L come from the words "first", "outer", "inner", "last", and are a memory device for helping you remember how to multiply horizontally, without having to write out the distribution like I did, and without dropping any terms. Here is what FOIL stands for:

That is, FOIL tells you to multiply the first terms in each of the parentheses, then multiply the two terms that are on the "outside" (furthest from each other), then the two terms that are on the "inside" (closest to each other), and then the last terms in each of the parentheses. In other words, using the previous example:

Use FOIL to simplify $\left(x +3\right)\left(x +2\right)$

Okay; the instructions specify the method I have to use, so here goes:

"first": $\left(x\right)\left(x\right)= x^2$

"outer": $\left(x\right)\left(2\right)=2x$

"inner": $\left(3\right)\left(x\right)=3x$

"last": $\left(3\right)\left(2\right)=6$

Adding the results of the four multiplications together, and combining the two "like" terms in the middle, I get:

$x^2 +2x +3x +6$

$x^2 +5x +6$

I prefer the FOIL to obtain product instead of the rule for the square of a binomial ${\left(x+a\right)}^2$. Since FOIL is used for all kind of product and it is very easy to use.