A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

imqwerty

  • one year ago

fun question :)

  • This Question is Closed
  1. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Let l and m be real numbers such that \[l \neq 0\] . Prove that not all the roots of \[lx^4 + mx^3 + x^2 + x+1 = 0\] can be real.

  2. sohailiftikhar
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    fun question huh ? O.o

  3. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 6

    \[lx^4 + mx^3 + x^2 + x+1 = l(x^2+ax+b)(x^2+cx+d)\] It is sufficient if we show the discriminant of one of those quadratic factors is less than \(0\)

  4. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yes @ganeshie8 :)

  5. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 6

    this is more better : \[x^4 + px^3 + qx^2 + qx+q = (x^2+ax+b)(x^2+cx+d)\]

  6. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i hav not read about such questions till now...these questions belong to the syllabus of which class? @ganeshie8

  7. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    It's a real mess!!

  8. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    class 11th :)

  9. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 6

    yeah lets try alternatives

  10. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    what if we take the derivative...would it be helpful?

  11. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    can we take derivative of 0?

  12. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    there is a very short method to solve this problem :)

  13. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    hint-if this equation is hard to work with then try to convert the equation :)

  14. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 6

    Let \(f(x)=lx^4 + mx^3 + x^2 + x+1 \) then \(\begin{align}f(1/x) &= \frac{l}{x^4} + \frac{m}{x^3}+\frac{1}{x^2}+\frac{1}{x}+1\\~\\ &=x^4(l+mx+x^2+x^3+x^4) \end{align}\) clearly if the polynomial \(f(1/x)\) has four real roots, then so does the polynomial, \(g(x)=l+mx+x^2+x^3+x^4\), and vice versa. Next consider the sum of squares of roots of \(g(x)\) : \(\sum a^2 = \left(\sum a\right)^2- 2\sum ab = (-1)^2-2(1)=-1\lt 0\). However the sum of squares of real numbers cannot be negative, so it follows that the roots of \(g(x)\) are not all real.

  15. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    correct @ganeshie8 :)

  16. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @ganeshie8 I don't get how \(f(1/x) = x^4(l+mx +x^2+x^3+x^4)\) . Please explain me.

  17. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    in this step he putted 1/x in the function f(x) :)

  18. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 6

    sorry, it is a typo, should be : \[f(1/x) = \frac{1}{x^4}(l+mx +x^2+x^3+x^4)\]

  19. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yes, now, it makes sense

  20. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 6

    thnks for catching :)

  21. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    ok lol i didn't notice that typo XD :)

  22. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @ganeshie8 Suggestion: change your nick to "genius8" :)

  23. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    ^

  24. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.