Question 1

Would it be considered more "simplified" by factoring out the multiple "2" making it:
 2 (3x^2y  -  3x^2  -  y  + 4xy + 2 )  ?  Is either way considered more "simplified" or are both ways  equally accepted?  Thanks.

Accepted Answer

Well, technically the way you typed in your question is the most simplified version the answer can get. The version that Sal had written at 2:20 isn't wrong, but it can be simplified, and that is the way most teachers and test markers will accept, especially for CXC exams, SAT exams, ACT exams, and any other exams. Some teachers may accept Sal's answer, but would most likely write a note next to the answer saying that it can be simplified even more than it is, or they might write the most simplified version, which is your answer.

Question 2

aren't the exponents typically in decreasing order? if so, then why is 8xy after -2y in his answer?

Accepted Answer

Well, those numbers have the same exponent, so it doesn't really matter. You don't _need_ to have numbers organized by their exponents, but doing that sometimes helps visual people.

Question 3

I have a question that says: (x + y)^2 - (9xy - 6x^2) = 7x^2 - 7xy + y^2
Can someone explain how they got this answer?

Accepted Answer

(x + y)² - (9xy - 6x²) = 7x² - 7xy + y²
Let's first expand the square of sum (x + y)².
(x + y)² = x² + 2xy + y² 
If you don't know how I got the above equation, you may want to watch some videos on squaring binomials but this is basically what I did:
(x + y)² = (x + y)(x + y) = x² + 2xy + y² (you distribute each term when multiplying).
So now we can substitute our expansion of (x + y)² into our original equation:
(x + y)² - (9xy - 6x²) = x² + 2xy + y² - (9xy - 6x²)
Now lets distribute the coefficient of -1 into (9xy - 6x²):
x² + 2xy + y² - (9xy - 6x²) = x² + 2xy + y² - 9xy + 6x²
Now we combine terms:
x² + 2xy + y² - 9xy + 6x² = 7x² + 2xy + y² - 9xy
7x² + 2xy + y² - 9xy = 7x² - 7xy + y²
So:
(x + y)² - (9xy - 6x²) = 7x² - 7xy + y²

Question 4

Usually teachers want the order of the simplified polynomials in order from greatest to lowest so how would I go about ordering an equation with more than 2 variables in one term or with different exponents like in this example?

Accepted Answer

Your teacher may have a preference, but in general, we order in terms of degrees of x, then followed by y and then z, though it does not matter the degree of y or z, as long as the highest degree of x is leftmost. Here are some examples: ²³⁴
1)
`xy² + x²y³ + x⁴ = x⁴ + x²y³ + y²` and sometimes further as `x(x³ + xy³ + y³)`
2)
`xy² + x²y³ + y⁴ = x²y³ + xy² + y⁴` and sometimes further as `y²(x²y + x + y²)`
3) 
`xy² + z²x²y³ + z³y⁴ = x²y³z² + xy² + y⁴z³` and sometimes further as `y²(x²yz² + x + y²z³)`

I hope that helps

Question 5

Why do you add numbers with the same exponents/variables together? For example, why would you add 5x^2 and 10x^2? Would this work for 5x^2 plus 10y^2?

Accepted Answer

you can add like terms because no matter what value of x is used, x^2 will always be the same, so 5x^2 + 10 x^2 = 15x^2 (say x=1, 5 + 10  = 15, x = 2 gives 20 + 40 = 60, etc.).  
You cannot combine 5x + 10x^2 because different values of x would give different results if you tried to somehow combine them.  Likewise, you cannot combine different variables such as 5x^2 + 10y^2 because you could change both x and y independently and get different results.  If you tried to say it was 15x^2y^2, if x = 1 and y = 1 or x=0 and y = 0, you would be fine since 5 + 10 = 15 and 0 + 0 = 0, but any other values would be incorrect.  Lets say x = 0 and y =1, you would end up with 0 + 10 = 0 which is a false statement.

Question 6

Do we need to take out the common factor 2 out from the expression? I see that Sal didn't take it out in the video so I am a bit confused. Thank you.

Accepted Answer

Yes, it can be simplified a bit more. Click the "Sort by" drop-down menu near the comment section, and choose "Top Voted." Someone has already asked your same question, and you will find a good answer in that comment thread. =)

Question 7

Sal put the 2y in front of the 8xy. Does it matter which order? If so, why? Any rules I should know?

Accepted Answer

No, in addition the order doesn't matter, whether it's a polynomial or not. But order keeps things less complicated, and it'll be easier to deal with ordered polynomials than a messy one, not to mention that you'll less likely make mistakes. Order can be really useful, especially in multiplying/dividing polynomials and other more complicated problems.

Question 8

This problem :  7(a+b)+ 4(a+b)- 5(a+b) is in my math book. They asked me to simplify.  I said that the answer was 6a+ 16b. I got it incorrect. The true answer is 6(a+b). Where did I go wrong? I can't figure it out. I used the distributive property and my problem looked like this 7a+7b+4a+4b-5a+5b. So then I added all the like terms with the variable a ( 7a, 4a, -5a), I then added the like terms with the variable b (7b, 4b, 5b). Ending up with 7a+4a-5a= 6a. And 7b+4b+5b= 16b, therefore resulting in 6a+16b. PLEASE HELP!

Accepted Answer

When you distribute the -5, you only distributed the negative to the a, not to both terms, so -5(a+b) = -5a - 5b, so you could have gotten 6a + 6b which is the same as 6(a+b).  Since you have a+b in common, you could take this as common factor to get (7 + 4 - 5)(a+b) to get their answer.

Course: Algebra (all content) > Unit 10

Adding polynomials: two variables (intro)

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