Multiplying Special Case Polynomials 06:00 minutes

Video Transcript

Transcript Multiplying Special Case Polynomials

GSquared and FoxyCooke are competing hackers. They're always trying to best each other. This time, they're trying to hack into a company’s security system. Whoever gets into the system first will be the winner of eternal fame and glory. GSquared is at the first security wall. To gain access to the first level, he has to simplify this expression, the sum of 'a' and 'b' squared, AKA the sum of 'a' and 'b' times the sum of 'a' and 'b'.

The binomials

GSquared knows a good strategy when he sees one. He uses an area model to find the product of the two binomials.To set up the area model, he divides a rectangle into 2 rows and 2 columns and labels each of the 4 sections with a term from the 2 binomials. Then he calculates the area of each section and writes it down in the corresponding section of the area model.

(a)(a) = a²,
(a)(b) = ab,
(b)(a) = ab,
(b)(b) = b².

Calculate the product of two binomials

He adds the four terms together, groups the like terms and finally, writes the expression in the standard form. The result? a² + 2ab + b². He knows he's got this, so he enters the password…Gosh darn it. That FoxyCooke got here first. GSquared was too slow. Does Foxy know a faster way to calculate the product of two binomials? She does know a faster way…

Simplify the expression

When Foxy saw the expression, she immediately recognized the pattern. She knows the square of a binomial sum is equal to a perfect square trinomial! GSquared reaches the wall of the 2nd level. Hoping to be faster than last time, he tries a different method to simplify the expression. He uses the FOIL method to simplify the difference of a and b times the difference of a and b.

The FOIL method

FOIL stands for first, outside, inside, and last. This is a mnemonic device used to simplify the product of two binomials. Let's try it out, but we have to go fast! First, multiply the first two terms, (a)(a) = a². Next, multiply the outer terms, (a)(-b) = -ab. Now the inside terms, (-b)(a). The product is '-ab'. Last, the last terms, (-b)(-b) = b².

The perfect square trinomial

Combine the like terms and... the simplified expression is a² - 2ab + b². He enters the password and...Although he used the FOIL method, he's foiled again by Foxy – she beat him a second time. How did she do it? Again Foxy recognized a pattern. This expression is the square of a binomial difference and she used another perfect square trinomial to crack the password in just seconds.

Multiplication problem

Finally the last wall... This time the task is a multiplication problem, 42 times 38. GSquared is a pro at multiplication, so he's not worried... Are you kidding me? That Foxy is a real fast fox. Look how she used the difference of two squares to solve the problem fast like lightning. When you have the expression of the sum of a and b times the difference of a and b, watch what happens when you apply the FOIL method the inner and outer products cancel each other out to leave the difference of two squares!

The DOTS expression

This is also known as a DOTS expression That Foxy, she applied DOTS to the number 42 and 38 rewriting them as 40 plus 2 and 40 minus to get a difference of two squares, then she simplified the exponents and calculated the difference to equal 1,596! She recognized the pattern of a difference of two squares!

Pay attention and you'll start to notice these patterned expressions too, just like Foxy. This summary Foxy used is super helpful to help you learn to recognize the polynomial patterns, so you can save time and be Foxy-fast. Foxy's the clear winner, for sure , but hold on... What’s going on here? It looks as though our two hackers got hacked…by their mom! It`s time for dinner.