What's the Formula for the Product of a Sum and a Difference?

A product is the result of multiplication
A sum is the result of addition
A difference is the result of subtraction
a and b are variables
a+b is the sum of a and b
a-b is the difference of a and b
FOIL out the left hand side to verify the formula
-ab and ab are opposites, so they cancel out
Since we got the right side of the formula when we FOILed, we know our formula is correct

1. A product is the result of multiplication
2. A sum is the result of addition
3. A difference is the result of subtraction
4. a and b are variables
5. Their sum is a+b
6. Their difference is a-b
1. We can make sure the formula is correct by multiplying out the left hand side
1. FOIL is a method we use to multiply two binomials
2. A binomial is a polynomial with two terms
1. Multiply the first terms together, and get a•a
2. Multiply the outer terms together, and get a(-b)
3. Multiply the inner terms together, and get b•a
4. Multiply the last terms together, and get b(-b)
1. Multiply out each term to simplify
1. Since -ab is negative and ab is positive, they are opposites
1. -ab+ab=0
2. This is the right hand side of our formula!
1. Since we got a²-b² when we multiplied the left hand side, our formula must be correct!
1. A product is the result of multiplication
2. A sum is the result of addition
3. A difference is the result of subtraction

Definition: Product of a Sum and a Difference