Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing
abtrehearn
Group Title
Suppose P(x) = x^3  a x^2 + b x  c has three distinct real roots, r1, r2, and r3. What cubic polynonial with 1 as its leading coefficient has r1^4, r2^4, and r3^4 as its roots?
 3 years ago
 3 years ago
abtrehearn Group Title
Suppose P(x) = x^3  a x^2 + b x  c has three distinct real roots, r1, r2, and r3. What cubic polynonial with 1 as its leading coefficient has r1^4, r2^4, and r3^4 as its roots?
 3 years ago
 3 years ago

This Question is Closed

jamesm Group TitleBest ResponseYou've already chosen the best response.0
(x  r1)(x  r2)(x  r3) = x^3  ax^2 + bx  c x^3  (r1 + r2 + r3)x^2 + (r1r2  r3r1  r3r2)x  r1r2r3 = x^3  ax^2 + bx  c thus r1 + r2 + r3 = a [1] r1r2  r3r1  r3r2 = b [2] r1r2r3 = c [3] (x  r1^4)(x  r2^4)(x  r3^4) = x^3  (r1^4 + r2^4 + r3^4)x^2 + (r1^4*r2^4  r3^4*r1^4  r3^4*r2^4)x  r1^4*r2^4*r3^4 We can see from equations [1], [2] and [3], if we raise both sides of [3] to the power of 4: r1^4 * r2^4 * r3^4 = c^4 so last term of the polynomial is c^4
 3 years ago

jamesm Group TitleBest ResponseYou've already chosen the best response.0
struggling to see how to express the other terms in terms of a, b and c
 3 years ago

abtrehearn Group TitleBest ResponseYou've already chosen the best response.0
Looks good so far.
 3 years ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
isn't the answer just P(x)=(xr1^4)(xr2^4)(xr3^4)
 3 years ago

abtrehearn Group TitleBest ResponseYou've already chosen the best response.0
That's right.
 3 years ago

abtrehearn Group TitleBest ResponseYou've already chosen the best response.0
The coefficients of the polynomial we're looking for are to be expressed in terms of a, b, and c.
 3 years ago

jamesm Group TitleBest ResponseYou've already chosen the best response.0
requires an algebraic trick that i can not see...give me some time :) i get a cubic in r3 that looks hard to solve
 3 years ago

jimmyrep Group TitleBest ResponseYou've already chosen the best response.1
r1 + r2 + r3 = a r1r2 + r1r3 + r2r3 = b r1r2r3 = c we need to get expressions for similar iidenties as the above replacing r1 by r1^4 etc in terms of a, b an c but that seems really dificult
 3 years ago

jimmyrep Group TitleBest ResponseYou've already chosen the best response.1
you need to find r1^4 + r2^4 + r3^4 in terms of the three identities and then substitute a, b and c and similarly for the other two identities  thats a daunting task
 3 years ago

jamesm Group TitleBest ResponseYou've already chosen the best response.0
tough one isn't it, jimmy?
 3 years ago

jimmyrep Group TitleBest ResponseYou've already chosen the best response.1
i'm trying to expand r1^4 + r2^4 + r3^4  4r1r2r3 (similar to what you do with a cubic) but its a real headache)
 3 years ago

jimmyrep Group TitleBest ResponseYou've already chosen the best response.1
best of luck!  must go  i have to take the wife shopping.
 3 years ago

jamesm Group TitleBest ResponseYou've already chosen the best response.0
no worries, have fun
 3 years ago

jimmyrep Group TitleBest ResponseYou've already chosen the best response.1
lol  u must be joking!  i'm spending money!
 3 years ago

jamesm Group TitleBest ResponseYou've already chosen the best response.0
i give up. do you have a solution abtrehearn?
 3 years ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
\[x^3x^2*(a^4+R_a)+x(b^4R_b)c^4\]
 3 years ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
lol \[R_a\] means the remaining part of a^4 \[R_b\] means the remaining part of b^4
 3 years ago

jamesm Group TitleBest ResponseYou've already chosen the best response.0
the remaining part?
 3 years ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
a^4=r1^4+r2^4+r3^4+....+some other stuff there is a minus sign in front of this so i need to add the remaining stuff back
 3 years ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
so there should be a negative in front of R_a
 3 years ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
since there is already a negative in front of that parenthesis
 3 years ago

jamesm Group TitleBest ResponseYou've already chosen the best response.0
i'm not sure what R_a represents? can you find the remaining two coefficients in terms of a, b and c?
 3 years ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
R_a is the ramaining part of a^4 remember a^4=(r1+r2+r3)^4=r1^4+r2^4+r3^4+alot of other gibberish this gibberish is R_a
 3 years ago

jamesm Group TitleBest ResponseYou've already chosen the best response.0
yes, but the gibberish is in terms of r1, r2 and r3
 3 years ago

myininaya Group TitleBest ResponseYou've already chosen the best response.0
(r1+r2+r3)^4=r1^4+r2^4+r3^4+R(x)
 3 years ago

abtrehearn Group TitleBest ResponseYou've already chosen the best response.0
One thing that comes in handy is this identity: \[r _{1}^{3} + r _{2}^{3} + r _{3}^{3}  3r _{1}r _{2}r _{3} = (r _{1} + r _{2} + r _{3})(r _{1}^{2} + r _{2}^{2} + r _{3}^{2}  r _{1} r _{2}  r _{1}r _{2}r _{2}r _{3})\]
 3 years ago

abtrehearn Group TitleBest ResponseYou've already chosen the best response.0
\[= (r _{1} + r _{2} + r _{3})(r _{1}^{2} + r _{2}^{2} + r _{3}^{2}  r _{1}r _{2}  r _{1}r _{3}  r _{2}r _{3})\]
 3 years ago

abtrehearn Group TitleBest ResponseYou've already chosen the best response.0
\[= a(r _{1}^{2} + r _{2}^{2} + r _{3}^{2}  b)\] \[= r _{1}^{3} + r _{2}^{3} + r _{3}^{3}  3c\] Another useful identity is \[\[(r _{1} + r _{2} + r _{3})^{2} = r _{1}^{2} + r _{2}^{2} + r _{3}^{2} + 2(r _{1}r _{2} + r _{1}r _{3} + r _{2}r _{3})\]
 3 years ago

abtrehearn Group TitleBest ResponseYou've already chosen the best response.0
This identity leads to \[a ^{2} = r _{1}^{2} + r _{2}^{2} + r _{3}^{2} + 2b.\]
 3 years ago

abtrehearn Group TitleBest ResponseYou've already chosen the best response.0
So a^2  2b can sub in for r1^2 + r2^2 + r3^2 in the cubic identity.
 3 years ago

abtrehearn Group TitleBest ResponseYou've already chosen the best response.0
If we have x^3 = a x^2  b x + c for x in {r1, r2, r3}, the same is true for x^4 = a x^3  b x^2 + c x = a(ax^2  bx + c) + b x^2 + c x = (a^2 + b) x^2 + (a b + c)
 3 years ago
See more questions >>>
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
 Engagement 19 Mad Hatter
 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.