Lesson: Factoring - Difference of Squares

Comment on Factoring - Difference of Squares

Hi Brent, thanks for all these great resources!
@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.
gmat-admin's picture

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 not essentially tied in with this video, but I have been scanning these vids trying to find a solution!
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.
gmat-admin's picture

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

gmat-admin's picture

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.
gmat-admin's picture

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-j-2-4-6-8-98-100-and-k-239753.html
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?
gmat-admin's picture

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?

gmat-admin's picture

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!
gmat-admin's picture

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
gmat-admin's picture

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/forum/is-x-2-y-2-a-positive-number-71000.html

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
gmat-admin's picture

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
gmat-admin's picture

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 right track. Thank you, that definitely helped.

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)
gmat-admin's picture

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 your solution to below question

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
gmat-admin's picture

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