Question

How to factor? 12x^6-27y^4.

I know to take out the three, but I don't know what to do from there. Any help? Thanks.

Answer 1

once you take out the 3, you have the difference of two squares

Answer 2

\[3(4x^6-9y^4)\] the inner piece is like \(a^2-b^2=(a+b)(a-b)\) with \(a=2x^3, b=3y^2\)

Answer 3

Ah, okay, I didn't know the formula. Thank you.

Answer 4

May I ask for the formula if it is a^2+b^2, @satellite73 ?

Answer 5

`a^2+b^2` is a sum of squares and can't be factored (assuming 'a' and 'b' have nothing in common)

Answer 6

I'm working on the problem now, just gimme a minute and we'll see if I got it right. :-)

Answer 7

I got 2x^5+3y^2)(2x^4-3y^2)

Answer 8

what @jim_thompson5910 said
you can factor the "difference of two squares" but not the "sum of two squares"

Answer 9

hmm no

Answer 10

@satellite73 How did you get 2x^3?

Answer 11

the exponent for the \(x\) term should be \(3\) in both cases

Answer 12

that is a good question

Answer 13

i visualized \(4x^6\) as \((2x^3)^2\)

Answer 14

I thought you just subtract. Since to the squared is to, you'd minus two from 6.

Answer 15

So you divide the exponent?

Answer 16

satellite73 is using the rule

\[\LARGE (x^a)^b = x^{a*b}\]

notice how the inner exponent 'a' is being multiplied with the outer exponent b

Answer 17

so with 4x^6, satellite73 broke up the 6 into 3*2 and used the rule above

Answer 18

Does my understanding of it? You divide?

Answer 19

*does it work ,too?

Answer 20

yeah 6/2 = 3

Answer 21

and 4/2 = 2

Answer 22

Does that work in general or was I just lucky there?

Answer 23

well again, notice how 6 factors into 3*2

we want the *2 portion so we can ultimately have something squared

so that's why we divide by 2 to figure out that other factor

Answer 24

if we had say x^10, we'd break it up like 5*2

how do we find 5? well we can say 10/2 = 5

Answer 25

Ah, okay, I am working on another problem right now. It would mean so much if you could stay for a bit to see if I got it right. :-)

Answer 26

This may take more for more as me as it involves big numbers. Thanks for your patience.

Answer 27

Finally got the factors. 28 and 28.

Answer 28

Hold on a bit.

Answer 29

(4x^2+7)^3

Answer 30

Is that right?

Answer 31

@satellite73 ?

Answer 32

I just realized I didn't give you the problem. XD

Answer 33

16x^4+56x^2+49

Answer 34

your answer is correct

Answer 35

Actually, according to the calculator, it's squared not cubed. :(

Answer 36

oh no it isn't the exponent should be 2

Answer 37

\[16x^4+56x^2+49\] is a "perfect square"

Answer 38

\[(a+b)^2=a^2+2ab+b^2\] in your case \(a=4x^2,b=7\)

Answer 39

Thank you. I'm fine for now, but I may be good. Thanks again for your time! :-)