Question

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?

Answer 1

(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

Answer 2

struggling to see how to express the other terms in terms of a, b and c

Answer 3

Looks good so far.

Answer 4

isn't the answer just
P(x)=(x-r1^4)(x-r2^4)(x-r3^4)

Answer 5

That's right.

Answer 6

The coefficients of the polynomial we're looking for are to be expressed in terms of a, b, and c.

Answer 7

requires an algebraic trick that i can not see...give me some time :)

i get a cubic in r3 that looks hard to solve

Answer 8

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

Answer 9

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

Answer 10

tough one isn't it, jimmy?

Answer 11

too right

Answer 12

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)

Answer 13

best of luck! - must go - i have to take the wife shopping.

Answer 14

no worries, have fun

Answer 15

lol - u must be joking! - i'm spending money!

Answer 16

i give up. do you have a solution abtrehearn?

Answer 17

\[x^3-x^2*(a^4+R_a)+x(b^4-R_b)-c^4\]

Answer 18

lol
\[R_a\] means the remaining part of a^4
\[R_b\] means the remaining part of b^4

Answer 19

the remaining part?

Answer 20

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

Answer 21

so there should be a negative in front of R_a

Answer 22

since there is already a negative in front of that parenthesis

Answer 23

i'm not sure what R_a represents?

can you find the remaining two coefficients in terms of a, b and c?

Answer 24

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

Answer 25

yes, but the gibberish is in terms of r1, r2 and r3

Answer 26

(r1+r2+r3)^4=r1^4+r2^4+r3^4+R(x)

Answer 27

One thing that comes in handy is this identity:
\[r _{1}^{3} + r _{2}^{3} + r _{3}^{3} - 3r _{1}r _{2}r _{3} = (r _{1} + r _{2} + r _{3})(r _{1}^{2} + r _{2}^{2} + r _{3}^{2} - r _{1} r _{2} - r _{1}r _{2}-r _{2}r _{3})\]

Answer 28

\[= (r _{1} + r _{2} + r _{3})(r _{1}^{2} + r _{2}^{2} + r _{3}^{2} - r _{1}r _{2} - r _{1}r _{3} - r _{2}r _{3})\]

Answer 29

\[= a(r _{1}^{2} + r _{2}^{2} + r _{3}^{2} - b)\]
\[= r _{1}^{3} + r _{2}^{3} + r _{3}^{3} - 3c\]
Another useful identity is
\[\[(r _{1} + r _{2} + r _{3})^{2} = r _{1}^{2} + r _{2}^{2} + r _{3}^{2} + 2(r _{1}r _{2} + r _{1}r _{3} + r _{2}r _{3})\]

Answer 30

This identity leads to
\[a ^{2} = r _{1}^{2} + r _{2}^{2} + r _{3}^{2} + 2b.\]

Answer 31

So a^2 - 2b can sub in for r1^2 + r2^2 + r3^2 in the cubic identity.

Answer 32

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)