There is x a complex number.
\(\displaystyle x^{3}=1\) so \(\displaystyle (x-1)(x^{2}+x+1)=0\)
I need to find the value of \(\displaystyle (1+x)(1+x^{2})(1+x^{3})(1+x^{4})(1+x^{5})(1+x^{6})\)
I solved the exercise but I didn't have an algorithm to follow.I just "played" with operations until I got something. (4 final answer which is correct)
I combined the first and the second terms and I multiplied them and I got 2(1+x^4)(1+x^5)(1+x^6)
The last term is 1+x^3*x^3 = 2
So I have 4(1+x^4)(1+x^5)
I multiplied the last two terms and I got 4(1+x^5+x^4+x^3*x^3) = 8 + 4(x^5+x^4) = 8 + 4x^4(x+1)
That "x+1" is -x^2 from x^3=1 so I got 8 - 4x^3 = 8-4 =4
My question is if there's another way to solve this exercise.I don't know, maybe it's something which I should notice.Something tricky, an "elegant" method.
Thanks!
\(\displaystyle x^{3}=1\) so \(\displaystyle (x-1)(x^{2}+x+1)=0\)
I need to find the value of \(\displaystyle (1+x)(1+x^{2})(1+x^{3})(1+x^{4})(1+x^{5})(1+x^{6})\)
I solved the exercise but I didn't have an algorithm to follow.I just "played" with operations until I got something. (4 final answer which is correct)
I combined the first and the second terms and I multiplied them and I got 2(1+x^4)(1+x^5)(1+x^6)
The last term is 1+x^3*x^3 = 2
So I have 4(1+x^4)(1+x^5)
I multiplied the last two terms and I got 4(1+x^5+x^4+x^3*x^3) = 8 + 4(x^5+x^4) = 8 + 4x^4(x+1)
That "x+1" is -x^2 from x^3=1 so I got 8 - 4x^3 = 8-4 =4
My question is if there's another way to solve this exercise.I don't know, maybe it's something which I should notice.Something tricky, an "elegant" method.
Thanks!