Find the value of \(\displaystyle{x}^{{{5}}}+{\frac{{{1}}}{{{x}^{{{5}}}}}}\) - question

Achatesopo8

Achatesopo8

Answered question

2022-04-07

Find the value of x5+1x5 - question about correctness of method

Answer & Explanation

enchantsyseq

enchantsyseq

Beginner2022-04-08Added 19 answers

Step 1
The fact is that
x+1x=1x2x+1=0
x=1±32
=1±i32C
In general, when proving equalities, we are always allowed to take square. We need to pay attention when we are solving an equation for x, in these case infact squaring both sides can produce some extra solutions which must be checked with respect to the original equation.
gil001q4wq

gil001q4wq

Beginner2022-04-09Added 11 answers

Step 1
If you think on x+1x=1 as x2x+1=0, a nicer fact is:
For any cuadratic equation ax2+bx+c=0 with roots x1, x2. If you define St=x1t+x2t for tR, this relation always hold:
aSt+1+bSt+cSt1=0
(You can think like changing xt for St).
Since always S0=2, and S1=x1+x2=bA, you can easy find S1. Later with some recursion you cand find S2, S3, S4 and finally S5=x15+x25.
This relation holds for all polynomials (even complexs one).

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