SOLUTION: If 𝛼, 𝛽, 𝛾 (where 𝛼, 𝛽, 𝛾 ≠ 0) are the roots of the equation 𝑥 3 + 𝑝𝑥^2 + 𝑞𝑥 + 𝑟 = 0, where 𝑝, 𝑞 and 𝑟 (≠ 0) are real numbers, ex

Algebra ->  Equations -> SOLUTION: If 𝛼, 𝛽, 𝛾 (where 𝛼, 𝛽, 𝛾 ≠ 0) are the roots of the equation 𝑥 3 + 𝑝𝑥^2 + 𝑞𝑥 + 𝑟 = 0, where 𝑝, 𝑞 and 𝑟 (≠ 0) are real numbers, ex      Log On


   

Question 1205171: If 𝛼, 𝛽, 𝛾 (where 𝛼, 𝛽, 𝛾 ≠ 0) are the roots of the equation 𝑥
3 + 𝑝𝑥^2 + 𝑞𝑥 + 𝑟 = 0, where 𝑝, 𝑞
and 𝑟 (≠ 0) are real numbers, express the following in terms of 𝑝, 𝑞 and 𝑟:
𝛼^2𝛽^2 + 𝛽^2𝛾^2 + 𝛼^2𝛾^2
Found 2 solutions by Bogz, ikleyn:
Answer by Bogz(13) About Me  (Show Source):
You can put this solution on YOUR website!
If alpha, beta, gamma are the roots of x%5E3%2Bpx%5E2+%2B+qx+%2Br+=+0, then we know that
alpha+%2B+beta+%2B+gamma+=+-p,
alpha%2Abeta+%2B+beta%2Agamma+%2B+alpha%2Agamma+=+q, and
alpha%2Abeta%2Agamma+=+-r.
The 2nd equation above gives
%28alpha%2Abeta+%2B+beta%2Agamma+%2B+alpha%2Agamma%29%5E2+=+q%5E2.
But the left side of the preceding equation expands to
.
Hence, .
Therefore,

Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.

In this problem, the given restrictions  𝛼,  𝛽,  𝛾 ≠ 0  and   𝑟 ≠ 0   are not necessary and do not matter.

The solution,  given by the other tutor,  works nicely without these restrictions,  too.


My conclusion:     So,  these restrictions can be omitted.
                           They are not necessary.