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.

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:
[tex](a+b)(a-b)[/tex]

Multiply the Fronts of both Equations:
[tex](a*a)[/tex]

Multiply the Outsides of both Equations:
[tex](a*a)+(a*-b)[/tex]

Multiply the Insides of both Equations:
[tex](a*a)+(a*-b)+(b*a)[/tex]

Multiply the Lasts of both Equations:
[tex](a*a)+(a*-b)+(b*a)+(b*-b)[/tex]

Simplify.
[tex]=a^2-ab+ba-b^2[/tex]

[tex]=a^2-b^2[/tex]


Alternative Method (My Prefered Method)
[tex](a+b)(a-b)=a(a+b)-b(a+b)[/tex]

Use Regular Distributive Property.
[tex]a(a+b)-b(a+b)[/tex]

[tex]a*a+a*b-b*a+-b*b[/tex]

Simplify.
[tex]a^2+ab-ba+-b^2[/tex]

[tex]a^2-b^2[/tex]

Hope this helps.
-Benjamin