## Cutiepo0 2 years ago How to factor (x^3 + 1)

1. Brittni0605

$(x^{3}+1)$

2. Cutiepo0

mmhm!

3. bandicoot12

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

4. Hero

$(a + b)^3 = (a + b)(a^2 - ab + b^2)$

5. Brittni0605

$(x+1)(x^{2}-x+1)$

6. Hero

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

7. bandicoot12

ya basically what they said

8. Cutiepo0

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

9. Hero

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

10. Cutiepo0

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

11. Hero

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

12. Hero

Remember, $$1 = 1 \times 1 \times 1 = 1^3$$

13. Cutiepo0

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

14. Hero

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

15. Hero

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

16. Cutiepo0

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

17. Hero

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

18. Hero

But in general, they will not be equal

19. Cutiepo0

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)

20. Cutiepo0

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

21. Cutiepo0

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

22. Hero

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

23. Hero

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.

24. Hero

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

25. Hero

The silence is deafening

26. Cutiepo0

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

27. Hero

Oh goodie :D