Question

How do you Factorise this polynomial eq x^3 - 8

Answer 1

8=(2)^3 therefore it is (x^3-2^3)= a^3-b^3

Answer 2

\[(a^3-b^3)=......(x^3-2^3)\]

Answer 3

actually
(a^3-b^3) = (a-b)(a^2+ab+b^2).

Answer 4

I think you can derive all of these from pascal's triangle

Answer 5

also (a-b)[(a^2+2ab+b^2)-ab]

Answer 6

(a-b)[(a+b)^2-ab]

Answer 7

@rubyangelgirl is it clear or you need to factorize it further ...lol

Answer 8

@rubyangelgirl It's actually a beautiful thing.\[a- b\implies a- b\\a^2 - b^2 \implies (a - b)(a + b)\\a^3 - b^3 \implies (a - b)(a^2 + ab + b^2)\\a^4 - b^4 \implies (a - b)(a^3 + a^2b + ab^2 + b^3) \]Does anyone notice a pattern? ;)

Answer 9

exactly...

Answer 10

this is pascals right?

Answer 11

Thanks guys I understand now :)

Answer 12

Not sure what you mean, zzr0ck3r.
The factorizations have two parts.
First part: \((a - b)\) stays everywhere.
Second part: If we are factoring \(a^n - b^n\), then the second part is \((a + b)^{n - 1}\) with maximum coefficient = 1.

Answer 13

Ahh, I jsut thought there was a way to get here using his triangle, but I dont really know what I'm talking about as I have never studied it.

Answer 14

For example, we have \(a^3 - b^3\).

First part: \((a - b)\)
Second: \((a + b)^2 \implies a^2 + 2ab + b^2\). But the maximum coefficient is 1... so we get the second part as \(a^2 + ab + b^2\).

Answer 15

ahh very cool

Answer 16

What if it's (x+1)^3 - (y-2)^3

Answer 17

That's already factorized. Isn't it?

Answer 18

And for all of you who have Mathematica, I wrote a little observation code for y'all using Manipulate. Paste this and run it:

Manipulate[Factor[a^n - b^n], {n, 1, 10, 1}]

Answer 19

@ParthKohli Yes it is but the question says to factorize it.

Answer 20

Oh, I got it. Use the difference of cubes formula again.

Answer 21

\[\large a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]In your question, \(a = x + 1\) and \(b = y - 2\).

Answer 22

Yes, but in the second do i have to expand the whole thing and then simplify ?

Answer 23

Yeah. :(

Answer 24

No...

Answer 25

First, plug in:\[\large a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]\[ \implies \large (x + 1)^3 - (y - 1)^3 \]\[\implies \large = ((x + 1) - (y - 1))((x + 1)^2 + (x + 1)(y - 1) + (y - 1)^2) \]

Answer 26

Just use technological help here.

Answer 27

Thanks :)