How do you FOIL # (x^3 – x^2)(x^3 + x^2)#?
FOIL is a mnemonic to help remember all of the combinations you need to multiply and add when finding the product of two binomials.
In our example, we find:
#(x^3-x^2)(x^3+x^2) = overbrace((x^3)(x^3))^"First"+overbrace((x^3)(x^2))^"Outside"+overbrace((-x^2)(x^3))^"Inside"+overbrace((-x^2)(x^2))^"Last"#
#color(white)((x^3-x^2)(x^3+x^2)) = x^6+color(red)(cancel(color(black)(x^5)))-color(red)(cancel(color(black)(x^5)))-x^4#
#color(white)((x^3-x^2)(x^3+x^2)) = x^6-x^4#
In fact, in this particular example, we might note that the multiplication takes the form
In general, we have:
#(A-B)(A+B) = A^2-B^2#
This is known as the difference of squares identity.
In our particular example:
#(x^3-x^2)(x^3+x^2) = (x^3)^2-(x^2)^2 = x^6-x^4#
FOIL is extremely useful in multiplying binomials. To be able to use it, let's understand what it means first:
FOIL stands for Firsts, Outsides, Insides, Lasts, meaning we multiply the first terms, outside terms, inside terms, and last terms, respectively.
Thus, we have:
The middle terms will cancel out (as expected, because the original problem is in the difference of squares format), and we get:
We can factor out an
the answer is
_ First Outer Inner __ Last