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use x^6 = 1+(x-1)*(1+x+x^2+x^3+x^4+x^5). Given (1+x+..+x^5) = 0 (x~=0) gives x^6 = 1 = exp(i*2*n*pi) . Hence x=exp(i*n*pi/3) where n= 1:5.This method gives same result but seems easier to use.
-1
That is how I did it to generate all of the roots of the equation.
x^4 + x^2 + 1 is always greater then 0, so, only x+1 can be equel to 0. So, x = -1.
Rational roots theorem to see that -1 is root. Synthetic division .... Child's play.
I directly put -1 and got the correct result without any calculation
3 odd exponents gave it away.
Much easier to multiply by x-1 to get x^6-1 =0 and x not equal to 1.
共通因数でくくる方法もあるけど、右辺を左辺に移項させて、両辺に(x-1)をかけると、x^6-1＝0となり、x＝1以外の複素数解になりますね。ここで複素平面上に半径1の円を書いて、ル·モアブルの定理より、中心角がπ/6の正6角形を書けば、自ずと分かりますね。
"As is obvious to the casual observer ..." was the phrase we used in my algebra class ... 50 years ago.
Excellent 👍👍👍👍❤🌹🇩🇿
It is a summ of geometric progression.
-1 (real value).
Or, Y = x^4 + x^3 + x^2 + x + 1so xy + 1 = 0 and x^5 + y = 0sub y = -x^5 into xy + 1 = 0 get x^6 = 1
X=-1
Easy enough by inspection.
Knew this IMMEDIATELY.
use x^6 = 1+(x-1)*(1+x+x^2+x^3+x^4+x^5). Given (1+x+..+x^5) = 0 (x~=0) gives x^6 = 1 = exp(i*2*n*pi) . Hence x=exp(i*n*pi/3) where n= 1:5.
This method gives same result but seems easier to use.
-1
That is how I did it to generate all of the roots of the equation.
x^4 + x^2 + 1 is always greater then 0, so, only x+1 can be equel to 0. So, x = -1.
Rational roots theorem to see that -1 is root. Synthetic division .... Child's play.
I directly put -1 and got the correct result without any calculation
3 odd exponents gave it away.
Much easier to multiply by x-1 to get x^6-1 =0 and x not equal to 1.
共通因数でくくる方法もあるけど、右辺を左辺に移項させて、両辺に(x-1)をかけると、x^6-1＝0となり、x＝1以外の複素数解になりますね。
ここで複素平面上に半径1の円を書いて、ル·モアブルの定理より、中心角がπ/6の正6角形を書けば、自ずと分かりますね。
"As is obvious to the casual observer ..." was the phrase we used in my algebra class ... 50 years ago.
Excellent 👍👍👍👍❤🌹🇩🇿
It is a summ of geometric progression.
-1 (real value).
Or,
Y = x^4 + x^3 + x^2 + x + 1
so xy + 1 = 0 and x^5 + y = 0
sub y = -x^5 into xy + 1 = 0 get x^6 = 1
X=-1
Easy enough by inspection.
Knew this IMMEDIATELY.