## Callisto 3 years ago Tutor question #2 Factor: x^4 -7x^2 +1 (conditions: pen, paper, no calculator, done in 2 minutes) **

1. maheshmeghwal9

May be we can take x^2 as "t" & then factor

2. maheshmeghwal9

I mean$(t^2-7t+1+5)-5$

3. Callisto

If my brain still works, Then it becomes (t-6)(t-1)-5 and then ..?

4. ParthKohli

I should learn this :/

5. maheshmeghwal9

ya it is not going to be factor like this:/ May be @satellite can help us:)

6. ParthKohli

It's @satellite73

7. ParthKohli

And he's offline

8. maheshmeghwal9

oops sorry lol

9. maheshmeghwal9

so we made only this much$(x^2-6)(x^2-1)-5.$ now what should we do of {-5}?

10. Callisto

I don't think we should do it in this way...

11. maheshmeghwal9

ya u r right @Callisto But I m thinking more on this.

12. bronzegoddess

you need to use synthetic division

13. Callisto

@bronzegoddess Can you further explain it?

14. maheshmeghwal9

x^2-6 @ParthKohli

15. Callisto

@ParthKohli It's not like that.... and you made mistakes there too...

16. ParthKohli

I deleted it before you noticed :P

17. maheshmeghwal9

lol

18. Callisto

I noticed before you deleted :P

19. maheshmeghwal9

ya i too

20. ParthKohli

This degree 4 is eating my head >.<

21. maheshmeghwal9

I m feeling giddy too

22. ParthKohli

@maheshmeghwal9 can you call Yogesh please?

23. maheshmeghwal9

but he is sitting aside of me

24. maheshmeghwal9

we 3 are all here:)

25. ParthKohli

Haha nice

26. bronzegoddess

okay, lets make a guess that (x-2) is a factor then we'll use synthetic division to actually se eif it one, its a trial and error method though: |dw:1339672516505:dw| so according to this division(x-2) isnt a factor because it gives us a remainder 11.

27. ParthKohli

But my wolfy says (x^2 - 3x + 1)(x^2 + 3x + 1)

28. Callisto

@bronzegoddess According to the word 'factor', I don't expect there is a remainder.

29. maheshmeghwal9

ya @Callisto is right: D

30. Callisto

@ParthKohli I said no wolf. I got that answer from wolf too. But that doesn't help you solve it because you cannot use calculator. Just pen and paper

31. ParthKohli

Hmm but I was telling bronzegoddess that this can be factored. I wanna understand too

32. ParthKohli

LANA THE PROFESSOR :D

33. Callisto

Not using synthetic division like that - yes :) Lana~~~~~~~~~~~~

34. bronzegoddess

@Callisto as I said it is a trail and error, and i was giving you an example, of course you have to start with finding the possible factors using descartes rule of signs!

35. maheshmeghwal9

We must want an exact & good method.

36. bronzegoddess

okay, let me do the whole thing again.

37. Callisto

@bronzegoddess If the factor is 2 terms like ax+b, I can do it. But in this case, it's not. So I don't know how. If you don't mind, can you teach me how to reach that answer?

38. maheshmeghwal9

very good @ajprincess

39. maheshmeghwal9

but how did she know about adding -3x^3 & something other. I want to hear from her.

40. maheshmeghwal9

ya but if we didn't know the answer, then what to do It is the exact problem:/

41. ajprincess

ya will show that in a few minutes time.

42. maheshmeghwal9

hmm..

43. bronzegoddess

first: number of possible roots is 4 because of the degree. second Descartes' Rule: number of +real roots = 1or zero, and -real roots= 1 or zero, imaginary=2 or 4 third: what are the possible roots; constant term/leading coefficient so $\pm1/\pm1$ therefore your 4 possible roots are -1,+1 and 2 imaginary numbers. fourth: synthetic division to see if -1 and +1 are actual roots; |dw:1339673466435:dw| but since it gives us a remainder then it isn't a factor(x+1) then you do the same with(x-1) and if it also give you a remainder then all your factors are imaginary. Thats how I learnt it.

44. Callisto

Do you know when to use that method?

45. Callisto

@bronzegoddess

46. ajprincess

we use trial and error method when we need to factorise expressions with greater power

47. bronzegoddess

when looking for factors or zeros of polynomials

48. Callisto

@bronzegoddess please take a look at this: http://en.wikipedia.org/wiki/Factor_theorem We can apply that method when one of the factors is in the form (x-k) Now, in this case, no factors are in the form (x-k). So, it cannot be applied in this case.

49. Callisto

Of course you can keep using that method for checking, but you can't get a factor, I think.

50. bronzegoddess

@Callisto (x-k) aka (x-1)=0 is x=1 which you'll end up using as the divisor in synthetic division.

51. Callisto

Okay, to be simple, if you can get a factor using that method, please tell me.

52. bronzegoddess

my step by step explanation from before doesn't make sense?

53. Callisto

Sorry, to be honest, it makes no sense to me in this case. I truly understand that method and have been using to to solve more than 50 questions - at least. But in this case, I don't think it can help me get a factor. Sorry if I offend you. But please read the things I posted, and I hope you understand that.

54. Callisto

And thank you for your time explaining that to me.

55. ajprincess

I agree @Callisto u cant get the answer in ths method. I tried with this method at first but didnt get the answr.

56. Callisto

*have been using that method to solve. Typo. Sorry

57. bronzegoddess

Ah so you already know synthetic division, I am not offended at all. I just wished to help but unfortunately it's not working. Sorry for causing confusion then.

58. Callisto

@bronzegoddess Thanks for your time spent here and willingness to help! I appreciate your efforts put here.

59. bronzegoddess

But there is of 4 factors: option1: 2 real(a+ and a -) and 2imaginary or option2: they are all imaginary isn't it?

60. maheshmeghwal9

May be of help @ some other time:)

61. Callisto

I just need real factors. The expression can be factorised, according to wolf

62. ajprincess

this sorts of questions can be solved without using calculators only by splitting the terms and by adding and removng additional terms. we will knw which terms to add and remove nly by workng many more questions of this type. I dnt thnk there is a specific way to solve this as far as I knw.

63. maheshmeghwal9

64. ParthKohli

I'm astonished. A question that looks as easy as simple factorization is confusing the polymaths.

65. maheshmeghwal9

66. bronzegoddess

when you find an answer please do inform me :) thank you!

67. ajprincess

Bt seems so useful @maheshmeghwal9.

68. maheshmeghwal9

thanx:) @ajprincess

69. ajprincess

yw.:)

70. maheshmeghwal9

If u find answer @Callisto Plz inform me:)

71. experimentX

The general way to solve quartic equation is to Ferrari's method http://www.proofwiki.org/wiki/Ferrari%27s_Method ... for this particular method, assuming it has two quadratic factors, $(x^2 + ax + b)(x^2 + cx +d) = x^4 - 7x^2 + 1 \\ \text{ or, } \\ x^4 + (a+c)x^3 + (b+d+ac)x^2 + (ad+bc)x + bd = x^4 - 7x^2 + 1 \\ \text{ Comparing coefficients } \\a+c = 0; \\b+d+ac=-7 \\ad+bc=0 \\ bd = 1$ Now solving the values these equations we get the coefficients http://www.wolframalpha.com/input/?i=c%2Ba%3D0%2Cb%2Bd%2Bac%3D-7%2Cad%2Bbc%3D0%2C+bd%3D1 Hence we have our quadratic factors http://www.wolframalpha.com/input/?i=factorize+x^4+-+7x^2+%2B1 The other (numerical) method would be Newton's method http://en.wikipedia.org/wiki/Newton's_method The best rule would be to solve this type of equation simply is view analytically the function f(x) = x^4 -7x^2 + 1 eg, f(x) = x^4 +7x^2 +1 does not have any real solutions Besides Descartes's rule sign change and other techniques would be helpful if we were able to guess it's solution (using remainder theorem) then also we can easily factor out using synthetic division

72. bronzegoddess

@experimentX that was what i was saying all along :) is there a way to find the imaginary solutions then?

73. experimentX

yes there is, Ferarri's method would be the general method ... check that link it finds out both real and imaginary roots ... (it's a lot messy though ... even for simple problems) Easy method is to reduce it into quadratic factors or, reduce into cubic (x-a)(x^3+bx^2+cx+d) = 0 and use Cardano's method <--- this is messy too (x^2+bx+c)(x^2+dx+e) = 0 <--- this is the best you can expect from that equation as you know the roots are imaginary if 4ac > b^2 for quadratic factors.

74. maheshmeghwal9

Hmm.... I heard of Ferrari today "A car also comes of help to maths{LOL}"

75. Callisto

Thanks