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.
Answer (1)
Use the FOIL method (First, Outer, Inner, Last) to expand ( a + b ) ( a − b ) .
Multiply the terms: First ( a × a = a 2 ), Outer ( a × − b = − ab ), Inner ( b × a = ab ), Last ( b × − b = − b 2 ).
Combine the terms: a 2 − ab + ab − b 2 .
Simplify to get the final answer: a 2 − b 2 .
Explanation
Understanding the Problem We are asked to use the FOIL method to multiply ( a + b ) ( a − b ) , and then describe a shortcut to find the product of ( a + b ) ( a − b ) without using FOIL.
Explaining the FOIL Method The FOIL method stands for First, Outer, Inner, Last. It's a technique used to multiply two binomials.
Applying the FOIL Method Let's apply the FOIL method to ( a + b ) ( a − b ) :
First: Multiply the first terms of each binomial: a × a = a 2 .
Outer: Multiply the outer terms of the binomials: a × − b = − ab .
Inner: Multiply the inner terms of the binomials: b × a = ab .
Last: Multiply the last terms of each binomial: b × − b = − b 2 .
Combining the Terms Now, combine all the terms we got from the FOIL method: a 2 − ab + ab − b 2
Simplifying the Expression Simplify the expression by combining like terms. Notice that − ab and + ab cancel each other out: a 2 − ab + ab − b 2 = a 2 − b 2
Describing the Shortcut The shortcut to find the product of ( a + b ) ( a − b ) without using FOIL is to recognize that this is a special product called the "difference of squares". The difference of squares formula is: ( a + b ) ( a − b ) = a 2 − b 2
Applying the Difference of Squares Formula So, instead of using the FOIL method, you can directly apply the difference of squares formula to get the same result: a 2 − b 2 .
Final Answer Therefore, the product of ( a + b ) ( a − b ) is a 2 − b 2 .
Examples
The difference of squares, ( a + b ) ( a − b ) = a 2 − b 2 , is a fundamental concept in algebra and has numerous real-world applications. For example, consider a rectangular garden that is being redesigned. Initially, the garden has sides of length a + b and a − b . The area of the original garden is ( a + b ) ( a − b ) . If we want to find the difference between the area of a square with side a and a square with side b , we can use the difference of squares formula to find that the area is a 2 − b 2 . This concept is also used in engineering to calculate areas and in physics to solve problems involving energy and momentum.
