Some polynomial question (1 Viewer)

[sasquatch]

The equation x³ + x² + 2 = 0 has the roots A, B, C (alpha, beta, gamma)

Evaluate:
a) A + B + C
b) A²+ B² + C²
c) A³+ B³ + C³
d) A⁴+ B⁴ + C⁴

I can do a, b and c, but not d. Could anybody help me out please?

 

[speedie]

a) A+B+C= -1
b) A^2+B^2+C^2= (A+B+C)^2 -2(AB +BC+AC)
= 1- 0
=1

c) sub roots into eqn: so A^3+A^2+2=0
similarly B^3+B^2+2=0
and C^3+C^2+2=0

now add all 3 eqns: A^3+B^3+C^3 +A^2+B^2+C^2+ 8=0
A^3+B^3+C^3+ 1+8=0
therefore A^3+B^3+C^3=-9

for d) do the same

ie. sub A B C into eqn, but x the whole thing by A/B/C

A^4+A^3+2A=0
B^4+B^3+2B=0
C^4+C^3+2C=0
A^4+B^4+C^4 -9 -2=0
A^4+B^4+C^4= 11

i dunno if i made any transcripts- but thats the method to do it

 

[sasquatch]

Yeah you made some errors, you did 2 + 2 + 2 = 8, where its supposed to be 6, and thus parts c and d are incorrect, but i see how you do it now.

All you have to do for part d, is use

A^3+A^2+2=0

but multiply both sides by A. To solve A^4 + B^4.......i cant be bothered... right?

 

[speedie]

yeh

 

[sasquatch]

Thanks alot!!