Chinese Middle School Math Olympiad | Evaluate the polynomial with a cubic equation

How to find the value of this polynomial x^97 +x^ 98 + ... +x^103 from a cubic equation? In this video, we'll discuss a way to use the given cubic equation and get the answer.

[prabhudasmandal]

The explanation that deducing x=1( +or -)wrongly from xpower 4 is noteworthy.thanks.

[أبوأنساليمني-مح]

Right, , , hi level method ..follow you from Yemen

[nasrullahhusnan]

Consider a geometric series 1, x, x², x³, ...
Let s(n)=x^(n-1) be the n-th element, and S(n) be sum of any n consecutives elements. Thus for k=1, 2, 3, ...


=[x^(k-1)](1+x+x²+x³)
=0 as it is given that x³+x²+x+1=0
Hence any sum of four consecutive elements is 0.
Note that a=1+x+x²+...+x¹⁰³ is sum of 104 elements, which can be grouped into 24 quartets of four consecutive elements. Thus a=0. While
b=1+x+x²+...+x⁹⁶
is sum of 97 elements, can be group into 24 quartets of 4 consecutive elements and one element. Counting from the right, the one element left is 1 --> b=1.
Therefore x⁹⁷+x⁹⁸+...+x¹⁰³=a-b=-1

[j.r.]

I handled this problem just by quickly solving the given cubic equation as x = -1. (Yes, it's a cubic, so there are 3 roots, but the other two are complex. I assumed we wouldn't be working with i as a base. But these complex solutions work too anyway.) Then, since the series has a very limited number of terms, I simply evaluated each one by looking at whether the exponent was odd or even. Ended up with one -1 left over, so that was the answer. This method was extremely easy and fast. If it's not legitimate, why not?