Answer to Question #116818 in Algebra for desmond

Question #116818
. Given that w denotes either one of the non-real roots of the equation z3 = 1, show
that
(a) 1 + w + w2 = 0, and
(b) the other non-real root is w2. Show that the non-real roots of the equation
1 −u u3 can be expressed in the form Aw and Bw2, where A and B are real
numbers, find A and B.
1
Expert's answer
2020-05-22T18:07:28-0400
"z^3=1""z^3-1=0""(z-1)(z^2+z+1)=0"

"z^2+ z+1=0"


"z={-1\\pm\\sqrt{1^2-4(1)^2}\\over2}={-1\\pm i\\sqrt{3}\\over2}"

Cube Root of Unity Value


"w_1=1,\\ real"

"w_2={-1- i\\sqrt{3}\\over2}, complex"

"w_3={-1+ i\\sqrt{3}\\over2}, \\ complex"

b)


"z^3-u^3=0"

"z^3-u^3=(z-u)(z^2+u z+u^2)"

Then


"(z-u)(z^2+u z+u^2)=0"

"z^2+u z+u^2=0"


"z={-u\\pm\\sqrt{u^2-4u^2}\\over2}=u({-1\\pm i\\sqrt{3}\\over2})"

"z_1=u w_1=u\\cdot1=1\\cdot u+i\\cdot0, \\ real"

"z_2=u w_2=u\\cdot{-1- i\\sqrt{3}\\over2}=-{1\\over 2}u-i\\cdot{u\\sqrt{3}\\over 2},\\ complex"

"z_3=u w_3=u\\cdot{-1+ i\\sqrt{3}\\over2}=-{1\\over 2}u+i\\cdot{u\\sqrt{3}\\over 2},\\ complex"


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