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.

Explanation:

Step1: Apply FOIL First terms

Multiply first terms of each binomial:
$(a)(a) = a^2$

Step2: Apply FOIL Outer terms

Multiply outer terms of the binomials:
$(a)(-b) = -ab$

Step3: Apply FOIL Inner terms

Multiply inner terms of the binomials:
$(b)(a) = ab$

Step4: Apply FOIL Last terms

Multiply last terms of each binomial:
$(b)(-b) = -b^2$

Step5: Sum all FOIL results

Combine the products and simplify:
$a^2 - ab + ab - b^2 = a^2 - b^2$

Step6: Identify the shortcut

Recognize the pattern of a difference of squares, which applies to binomials of the form $(x+y)(x-y)$.

Answer:

Using FOIL:

1. First: $a \times a = a^2$
2. Outer: $a \times (-b) = -ab$
3. Inner: $b \times a = ab$
4. Last: $b \times (-b) = -b^2$

Summing these gives $a^2 - ab + ab - b^2 = a^2 - b^2$.

Shortcut: This is a difference of squares. For any two terms $a$ and $b$, the product $(a+b)(a-b)$ is always equal to $a^2 - b^2$, so you can skip FOIL and directly write the difference of the squares of the two terms.
Final product: $\boldsymbol{a^2 - b^2}$