Question

Use FOIL to explain how to find the product of (a + b)(a − b). Then describe a shortcut that you could use to get this product without using FOIL.

Answer 1

Greetings!

FOIL stands for:

Front
Outside
Inside
Last

This tells which terms to multiply when using the Distributive Property.
(NOTE: Only applicable with 2-term Polynomials)

For Example:
$(a+b)(a-b)$

Multiply the Fronts of both Equations:
$(a*a)$

Multiply the Outsides of both Equations:
$(a*a)+(a*-b)$

Multiply the Insides of both Equations:
$(a*a)+(a*-b)+(b*a)$

Multiply the Lasts of both Equations:
$(a*a)+(a*-b)+(b*a)+(b*-b)$

Simplify.
$=a^2-ab+ba-b^2$

$=a^2-b^2$


Alternative Method (My Prefered Method)
$(a+b)(a-b)=a(a+b)-b(a+b)$

Use Regular Distributive Property.
$a(a+b)-b(a+b)$

$a*a+a*b-b*a+-b*b$

Simplify.
$a^2+ab-ba+-b^2$

$a^2-b^2$

Hope this helps.
-Benjamin

Answer 2

Multiply the first, outside, inside, and last terms of the binomials.

Multiply a times a, a times -b, b times a, and b times -b.

This is the same as just squaring a and squaring b.

A shortcut is to add a squared and b squared.