How do you FOIL (x^3 – x^2)(x^3 + x^2)?

3 Answers
Mar 3, 2018

(x^3-x^2)(x^3+x^2) = x^6-x^4

Explanation:

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 (A-B)(A+B) with A=x^3 and B=x^2.

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

Mar 3, 2018

x^6-x^4, or alternatively x^2(x^3-x^2) (factored form)

Explanation:

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.

In (x^3-x^2)(x^3+x^2), our terms are as follows:

  • Firsts (x^3*x^3)= x^6
  • Outsides (x^3*x^2)= x^5
  • Insides (-x^2*x^3)= -x^5
  • Lasts (-x^2*x^2)= -x^4

Thus, we have:

x^6+x^5-x^5-x^4

The middle terms will cancel out (as expected, because the original problem is in the difference of squares format), and we get:

x^6-x^4

We can factor out an x^2 (which is essentially dividing by x^2). We get:

x^2(x^3-x^2)

Mar 3, 2018

the answer is x^6-x^4

Explanation:

(x^3-x^2)(x^3+x^2)=

=x^3x^3 + x^3x^2 - x^2x^3 - x^2x^2=
_ First Outer Inner __ Last
=x^6-x^4