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### Course: Digital SAT Math>Unit 4

Lesson 1: Factoring quadratic and polynomial expressions: foundations

# Structure in expressions — Harder example

Watch Sal work through a harder Structure in expressions problem.

## Want to join the conversation?

• What is Sal talking about
• I'm going to presume you know your multiplication table. If you don't... good luck I guess? Also you need to memorise the formulae of factorisation, good luck with that

Anyways the first step is substitution. We know that a is x^2 + y^2 and b is xy so the given equation becomes 9(x^2 + y^2) - 18 xy, or expanded and rearranged in descenting order of x's exponentials, 9x^2 - 18xy + 9y^2

Now if you remember the formula for expansion/factorisation, you'll be able to see that this expression you just calculated looks like a^2 - 2ab + b^2 = (a-b)^2. Now 3*3 = 9 and 3*3*2 = 18. So the answer is (3x-3y)^2

Did I already say that you need to learn the multiplication table by heart? It's really, really important!
• ok so we just forget about the middle number
• we didnt really forget about it. we used it to determine that we were dealing with a 3x 3y perfect square. when you multiply it out its the same
• Isn't -1 squared, +1. In the video he changed the form of the original equation to the new one with -1 squared; even though we need -1 as the result. So is this appropriate?
• Short Version \/
To address your first statement, (-1) squared would indeed equal 1. However, you are confusing (-1) squared with -(1) squared.

Long winded, rambling version \/
This confusion is a result of a devious property of exponents, as when you look at, say -2^2; you would think that the answer would be 4. However, -2 can be factored in a not so obvious manner into -1*2. Then, the exponent is applied to the positive two, leading to 4, which is then multiplied by negative one to become negative 4. The reason the exponent is applied before the multiplication of the -1 is because of the order of operations, parenthesis, exponents, multiplication/division and finally addition/subtraction. As exponents are before multiplication, and since -2^2 is also -1*2^2, thus the answer of the example is -4. If, however, parenthesis are present around the -2, then you will get 4.

• hey i am getting confused what is the difference between 3rd choice and the 4th one? it seems both of them can work
• The third one has an a squared rather than an a
• can someone explain why he ignored the -2(3x)(3y)?
• the first one seems to work tho
• At Sal says that it a difference of squares, which rules out option number one.
• bro just yAPPED
'
• wait how is 9x^2 equal to (3x)^2 ? I’m so confused 😐
• When we have exponents, we have to distribute them to each factor that we're dealing with. This is similar to when you are multiplying things, you distribute to every term ( 2*(3+x) = 2*3 + 2*x ). If you were "factoring" out an exponent of 2 from 9x^2, you would have to take it out of every term:
9x^2 = ( sqrt(9) * sqrt(x^2) )^2
= (3 * x)^2