Question

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

Answer 1

`(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`

Answer 2

`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)`

Answer 3

the answer is `x^6`-`x^4`

Explanation:

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

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