Question

How to factor (x^3 + 1)

Answer 1

\[(x^{3}+1)\]

Answer 2

mmhm!

Answer 3

no, (x+1)(x^2-x+1)

Answer 4

\[(a + b)^3 = (a + b)(a^2 - ab + b^2)\]

Answer 5

\[(x+1)(x^{2}-x+1)\]

Answer 6

That's the sum of cubes formula I posted above.

Answer 7

ya basically what they said

Answer 8

ok but I just established that (x^3+1) and (x+1)^3 weren't the same thing :S

Answer 9

No, they are not. One is simply x cubed plus one. The other is \((x + 1) \times (x + 1) \times (x + 1)\)

Answer 10

right, so then how can I use the sum of cubes formula? b/c isn't that for (a+b)3

Answer 11

Because
\((x^3 + 1) = (x^3 + 1^3)\)

Answer 12

Remember, \(1 = 1 \times 1 \times 1 = 1^3 \)

Answer 13

= (x + 1) ^3 ?? but wouldnt that make it the same??

Answer 14

No, I just explained to you earlier that \((x^3 + 1) \ne (x + 1)^3\)

Answer 15

\((x^3 + 1) = (x^3 + 1^3)\)

\((x + 1)^3 = (x + 1)(x + 1)(x + 1)\)

Answer 16

ok, so (a^3 + b^3) doesn't equal (a+b)^3, ever right? Sorry, just trying to wrap my head around it

Answer 17

The only way is if x = 0 or x = 1

Answer 18

But in general, they will not be equal

Answer 19

ok, so now though with the sum of cubes formula, the whole point was to find the zeroes of the equation f(x)= (x^3 + 1)(x-17)

Answer 20

and the answer is x=17 and x = -1

Answer 21

so how do I get from the sum of cubes to -1

Answer 22

Basically, the question is asking:

What x values would make \((x^3 + 1)(x - 17) = 0\)

You know that anything times zero equals zero.

So when you expand \((x^3 + 1)\) you get \((x + 1)(x^2 - x + 1)\)

Thus \((x^3 + 1)(x - 17) = (x + 1)(x^2 - x + 1)(x - 17) = 0\)

Answer 23

If you look closely you'll realize that if (x + 1) = 0 or if (x - 17) = 0, then the product of the whole thing will be zero. There's only one way to make (x + 1) = 0
There's only one way to make (x - 17) = 0.

Answer 24

What I say doesn't matter if you don't understand anything.

Answer 25

The silence is deafening

Answer 26

Oh, I got it, thanks a lot for your help with this :) b/c (x2−x+1) doesn't have any roots, so it's just the other two roots

Answer 27

Oh goodie :D