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Comment on Factoring - Difference of Squares
Hi Brent, thanks for all
@3.38, you showed x^2+81 and you factored it as 1(x^2+81),but isn't this a "Square of a Sum" and it can be written as (x+9)^2 = x^2+18x+81?
Would you please explain. Many thanks.
You are correct in that (x +9
You are correct in that (x +9)² = x² + 18x + 81
So, for example, if we want to factor the expression x² + 18x + 81, we can write: x² + 18x + 81 = (x+9)(x+9) = (x+9)²
However, in the video, we are trying to find a way to factor x² + 81. That is we want to write x² + 81 as the PRODUCT of two expressions. In other words, we want to write x² + 81 = (something)(something)
The ONLY way to do this is as follows x² + 81 = (1)(x² + 81)
Hi Brent, this question is
So I was on one of the gmat forums, and a member factored 6^4 + 3^4 as 3^4(2^4 +1). Would that not yield 6^8 + 3^4 since you add the exponents when multiplying through the brackets? Sorry if this sounds like a novice question, haven't attempted maths in a long time!
Cheers.
What you need here is the
What you need here is the Combining Bases Law. It can be found at 3:20 of https://www.gmatprepnow.com/module/gmat-powers-and-roots/video/1029
This law says that, if we have the same EXPONENTS in a product, we can combine the bases.
That is, (a^n)(b^n) = (ab)^n
So, (3^4)(2^4) = (3 x 2)^4 = 6^4
The rule you're referring to is correct (when we multiply two powers with the same BASE, then we add the exponents). However, in the case of (3^4)(2^4), we do not have the same bases, so we can't use that rule.
Hi Brent,
Grateful to let me have your answer for the following question please:
X^8 - Y^8 =
A.(X^4−Y^4)^2
B.(X^4 + Y^4)(X^2 + Y^2)(X + Y)(X − Y)
C.(X^6 + Y^2)(X^2 − Y^6)
D.(X^4 − Y^4)(X^2 − Y^2)(X − Y)(X + Y)
E.(X^2 − Y^2)^4
First, we need to recognize
First, we need to recognize that X^8 - Y^8 is a difference of squares, because X^8 = (X^4)(X^4) = (X^4)^2 and Y^8 = (Y^4)(Y^4) = (Y^4)^2
So, we can write: X^8 - Y^8 = (X^4 + Y^4)(X^4 - Y^4)
Then we must recognize that (X^4 - Y^4) is a difference of squares, which can be factored as (X^2 + Y^2)(X^2 - Y^2)
So, we get: X^8 - Y^8 = (X^4 + Y^4)(X^4 - Y^4)
= (X^4 + Y^4)(X^2 + Y^2)(X^2 - Y^2)
Finally, (X^2 - Y^2) is a difference of squares, which can be factored as (X + Y)(X − Y)
So, we get: X^8 - Y^8 = (X^4 + Y^4)(X^4 - Y^4)
= (X^4 + Y^4)(X^2 + Y^2)(X^2 - Y^2)
= (X^4 + Y^4)(X^2 + Y^2)(X + Y)(X − Y)
Answer: B
Does that help?
Cheers,
Brent
Hi Brent,
Link to Question: https://gmatclub.com/forum/1-220972.html
Could you please explain the FOIL method used in your explanation:
So,
Line 1:(1+√3+√5)² - (√3+√5)² = [(1+√3+√5) + (√3+√5)][(1+√3+√5) - (√3+√5)]
Line 2:= [1+2√3+2√5][1]
Can you explain how you go from simplifying line 1 to line 2? (I use the term "line" to distinguish the 2 steps).
During test day, would this not take a substantial amount of time to use the FOIL method on "[(1+√3+√5) + (√3+√5)][(1+√3+√5) - (√3+√5)]".
Thank you.
Link: https://gmatclub.com
Link: https://gmatclub.com/forum/1-220972.html
We definitely want to avoid using FOIL to expand [(1+√3+√5) + (√3+√5)][(1+√3+√5) - (√3+√5)]
Notice that, within each set of square brackets, we can do a lot of simplifying.
ASIDE: I noticed my original solution could have used more brackets, so I just added them.
Take the 2nd set of square brackets: [(1+√3+√5) - (√3+√5)]
We have √3 and then we subtract √3, leaving us with zero √3's
Likewise, we have √5 and then we subtract √5, leaving us with zero √5's
So, the 2nd set of square brackets simplifies to 1
That is: [(1+√3+√5) - (√3+√5)] = 1
We can also simplify the 1st set of square brackets: [(1+√3+√5) + (√3+√5)]
We have √3 and then we add another √3, giving us 2√3
Likewise, we have √5 and then we add another √5, giving us 2√5
We get: [(1+√3+√5) + (√3+√5)] = 1 + 2√3 + 2√5
Cheers,
Brent
https://gmatclub.com/forum/if
In this question sir i did like this:
1^2 - 2^2 = (1-2)(2+1) = -3
3^2 - 4^2 = (3-4)(3+4)= -7
and pattern continues only option c follows this pattern hence c
Sir is this approach correct?
Question link: https:/
Question link: https://gmatclub.com/forum/if-j-2-4-6-8-98-100-and-k-239753.html
Yes, I THINK that looks valid.
Can you elaborate on your solution?
Cheers,
Brent
Hey Brent
Referring to this Q: https://www.beatthegmat.com/if-b-is-greater-than-1-which-of-the-following-must-be-t298556.html
I substituted 1.1 for the value of b and tested all answers. Both D and E yielded negative solutions. Did I make an error in my calculations?
Question link: https://www
Question link: https://www.beatthegmat.com/if-b-is-greater-than-1-which-of-the-followin...
No, you didn't make an error.
The key word here is the word MUST, as in "Which of the following MUST be negative?"
This is the same as asking "Which of the following is negative FOR ALL VALUES OF B?"
So, when you plugged in b = 1.1, you found that answer choices A, B and C are positive.
So, you can eliminate those answer choices.
At this point, all we can really say is that answer choices D and E may SOMETIMES yield a negative output. However, we want to know which answer choice ALWAYS yields a negative output.
So, from here, we must test another value of b
If b > 1, b COULD equal 3. Let's plug b = 3 into our two remaining answer choices:
D. (2 - b)/(1 - b) = (2 - 3)/(1 - 3) = (-1)/(-2) = 1/2 POSITIVE
E. (1 - b²)/b = (1 - 3²)/3 = -8/2 NEGATIVE
Since answer choice D doesn't ALWAYS evaluate to be negative, we can eliminate it.
So, by the process of elimination, the correct answer is E.
This means (1 - b²)/b MUST be negative for ALL VALUES of b such that b > 1
Cheers,
Brent
Hi Sir,
Can you help me with this question, I got stuck on this one from your resources.
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then 12−22+32−42+52−62+.....+972−982+992−1002=12−22+32−42+52−62+.....+972−982+992−1002=
A) J² - K²
B) -50(J² - K²)
C) -K - J
D) K² - J²
E) (-J - K)²
Thank You!
Here's my full solution:
Here's my full solution: https://gmatclub.com/forum/if-j-2-4-6-8-98-100-and-k-239753.html#p1850079
Hi Brent
Here I tried a different approach.
https://gmatclub.com/forum/if-j-2-4-6-8-98-100-and-k-239753.html
I took a sample of the series of J and K, bearing in mind its conditions. Thus, in this example: J will be= 2,4 and K will be = 1,3.
1^2-2^2+3^2-4^2 equals 10. which is the sum of J+K. Remember J is 2+4 and K 1+3. Looking in the answer list there was no such answer as J+K, but -J-K is. which is, indeed, exactly the same. How do you see this approach? Best regards.
Alejandro
Question link: https:/
Question link: https://gmatclub.com/forum/if-j-2-4-6-8-98-100-and-k-239753.html
Great approach! It's basically the same as the INPUT-OUTPUT approach you'll learn about later here: https://www.gmatprepnow.com/module/gmat-word-problems/video/933
ASIDE: 1² - 2² + 3² - 4² = -10 (not 10)
So, -J - K works perfectly.
Cheers,
Brent
LINK: https://gmatclub.com
Statements 1 and 2 combined
Statement 1 tells us that (x - y) is POSITIVE
Statement 2 tells us that (x + y) is POSITIVE
So, (x + y)(x - y) = (POSITIVE)(POSITIVE) = POSITIVE
The answer to the target question is YES, x² – y² IS a positive number
Since we can answer the target question with certainty, the combined statements are SUFFICIENT
Answer: C
BUT
St1:
Case a: (x+y) = -ive
Case b: (x+y) = +ive
St2:
Case a: (x-y) = -ive
Case b: (x-y) = +ive
Thus, Why did we only used +ive results ie combination of Case A’s from both St1 & St2 to conclude a +ive result as:
(x+y)(x-y) = +ive ?
(+)(+) = (+) —> Okay
Whereas, we could use Case A of St1 & Case B of St2 that +ive & -ive respectively.
So, (x+y)(x-y) = +ive ?
(+) (-) =/= +ive —> Not Okay Thus Insufficient.
Therefore, Answer E —> Inconclusive/ambiguous even when combined
Question link: https:/
Question link: https://gmatclub.com/forum/is-x-2-y-2-a-positive-number-71000.html#p2424401
When we COMBINE both statements, we must consider two things as 100% true:
(x - y) is POSITIVE
(x + y) is POSITIVE
So, when analyzing the combined statements, we can't consider scenarios that don't satisfy the information provided in the statements.
For example, we can't consider a case in which (x + y) is POSITIVE and (x - y) is NEGATIVE, since statement 1 directly tells us that (x - y) is POSITIVE.
Does that help?
Hi Brent,
I'm having a bit of trouble with the problem below. I sort of understand it. It looks like the problem is testing the concept of difference of squares.
https://gmatclub.com/forum/if-a-and-b-are-different-values-and-a-b-a-b-then-in-terms-of-188100.html
Here's my full solution:
Here's my full solution: https://gmatclub.com/forum/if-a-and-b-are-different-values-and-a-b-a-b-t...
Please let me know if this helps
That is perfect. I was on the
Hi Brent,
Can you help me with solution for below question
https://gmatclub.com/forum/if-a-b-and-c-are-constants-a-b-c-and-x-3-x-x-a-x-b-167671.html
I am stuck at step x(x+1)(x-1)=(x-a)(x-b)(x-c)
You're almost there!
You're almost there!
We can rewrite x(x+1)(x-1) as (x - 0)(x - (-1))(x - 1)
So we get: (x - 0)(x - (-1))(x - 1) = (x - a)(x - b)(x - c)
This means, a = 0, b = -1 and x = 1
Here's my full solution: https://gmatclub.com/forum/if-a-b-and-c-are-constants-a-b-c-and-x-3-x-x-...
Oh now I get it...... thanks.
Hi Brent, I couldn’t find
https://gmatclub.com/forum/if-x-u-2-v-2-y-2uv-and-z-u-2-v-2-and-if-x-163805.html
Can you help me with this
Here it is: https://gmatclub
Here it is: https://gmatclub.com/forum/if-x-u-2-v-2-y-2uv-and-z-u-2-v-2-and-if-x-163... (it was on page 2)