## evy15 2 years ago Find all the zeros of the eq. 2x^4-5x^3+53x^2-125x+75=0

1. mathmate

This one can be factorized, so you will need to use the following tools: Descartes rule of signs, factorization theorem, fundamental theorem of algebra Are any or some of these famililar to you? Also, this may indicate that you or your teacher have a typo in the other problem. Please re-check for typos on the other problem.

2. mathmate

To make things simpler, when the sum of coefficients = 0, it means that (x-1) is a factor.

3. evy15

no I dnt know those methods

4. mathmate

Descartes rule of signs tells us that we have either 4 real positive roots, or two real positive roots and two complex roots, or 4 complex roots.

5. evy15

ok

6. mathmate

http://en.wikipedia.org/wiki/Descartes'_rule_of_signs http://en.wikipedia.org/wiki/Factor_theorem http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra Since you have been missing school for at least one day, it may well be advisable to catch up with some of these topics. Hope you got notes from your friends though.

7. mathmate

So is (x-1) a factor to the equation?

8. evy15

yes

9. mathmate

So you can do a synthetic division, or a long division to reduce the quartic to a cubic, i.e. find (2*x^4-5*x^3+53*x^2+(-125)*x+75)/(x-1)=?

10. evy15

2x^3-3x^2+50x-75

11. mathmate

Good, now we can rule out the third case (4 complex roots).

12. evy15

ok

13. mathmate

Using the factor theorem, you will need to try factors of the form (ax+b) where a=1 or 2 (from the leading coefficient 2), and b=+/- all possible factors of 75, which are 1,3,5,15,25,75, i.e. 2*2*6=24 possibilities. Once you find that, you can again do the synthetic division or long division to reduce the cubic to a quadratic.

14. evy15

what do I divide it by

15. mathmate

When you find another root (i.e. another factor) of the cubic.

16. evy15

oh ok, can I use the calculator for that

17. KingGeorge

Sorry to bust in here, but $$2x^3-3x^2+50x-75$$ is relatively easily factored by grouping, and thus we can skip the whole guess and check portion of finding roots.

18. mathmate

In fact, Descartes rule of sign says we don't have to try factors of the form (a+b), only those of the form (a-b) because all real roots are positive.

19. mathmate

So that reduces to 12 possibilities only.

20. mathmate

Very true, thanks KingGeorge!

21. KingGeorge

You group like so $2x^3-3x^2+50x-75=x^2(2x-3)+25(2x-3)$Can evy15 finish the factoring from here?

22. mathmate

@evy15 are you able to complete the solution from here?

23. evy15

can you guide me to what to do next

24. evy15

ok hold on

25. evy15

x=i+-5 and 3/2

26. evy15

i mean =-5i

27. mathmate

Good job, (+\- 5i). Now you have all 4 zeroes!

28. mathmate

Can you enumerate the four zeroes?

29. evy15

so its 1,+/-5i, and 3/2?

30. evy15

or-1

31. mathmate

Excellent! There you are! Not -1. Your list is good.

32. mathmate

Big thanks to KingGeorge who cut the work in half!

33. evy15

yes thank you guys so much

34. mathmate

yw! :)

35. KingGeorge

As an addendum, it's possible to factor the original function by grouping if you're clever. \begin{aligned} 2x^4-5x^3+53x^2-125x+75&=2x^4-5x^3+3x^2+50x^2-125x+75 \\ &=x^2(2x^2-5x+3)+25(2x^2-5x+3) \\ &=(x^2+25)(2x^2-5x+3)\\ &=(x^2+25)(x-1)(2x-3) \end{aligned}

36. KingGeorge

But this is not an obvious, nor the easiest, way to do it.

37. mathmate

Yep, I will look more carefully for these cases in the future.

38. evy15

can i do the same with x^5-3x^4-24x^3-72x^2+12x=12

39. evy15

sorry wrote the wrong one

40. evy15

x^5-3x^4-24x^3-72x^2-25x+75=0

41. KingGeorge

That can definitely be factored by grouping. However, instead of having two groups like I did before, we'll have three groups.

42. KingGeorge

Your first group will be $x^4(x-3).$Knowing this, can you give me the rest of the factorization?

43. evy15

x^4(x-3)-24x^2(x-3)-25(x-3)

44. KingGeorge

Perfect. And can you finish the rest of the factorization?

45. evy15

would it be (x^4-24x^2-25) and (x-3)?

46. KingGeorge

Right again. Now you just need to factor $$x^4-24x^2-25$$. You can do this by pretending it's a quadratic equation, and not a quartic. So how does $$x^2-24x-25$$ factor?

47. evy15

x-25 and x+1

48. KingGeorge

Perfect. So that means $$x^4-24x^2-25=(x^2-25)(x^2+1)$$. Hence, $x^5-3x^4-24x^3-72x^2-25x+75=(x^2-25)(x^2+1)(x-3)$From here, you just have some sums/differences of squares, and should be able to figure out the roots. I've got to go now, but you seem to be catching the hang of this!