Evaluating Polynomial Expressions | Math Olympiads

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[rex_yourbud]

Solution to original equation is ( -w², -w )
Put any of these in the expression to be evaluated
(-w)⁷ + (-w)⁵ = -w -w² = -(w+w²) = -(-1) = 1
things used :
w³ = 1
and 1 + w + w² = 0
w is the complex cube root of unity
Re(w) = -1/2 and Im(w) = sqrt(3)/2

[chaosredefined]

x - x^2 = 1
x^2 = x - 1

Square both sides
x^4 = x^2 - 2x + 1

Substitute in x^2 = x - 1
x^4 = (x - 1) - 2x + 1
x^4 = -x

Multiply both sides by x
x^5 = -x^2

Multiply by x^2
x^7 = -x^4

Substitute in x^4 = -x
x^7 = x

Note that x^7 = x and x^5 = -x^2. Substituting these into our goal equation:
x^7 + x^5 = x - x^2 = 1

[Hexa]

From Bangladesh.
Sir you are a legend.🥰🥰

[tanayshah]

if we know x³ is -1, why dont we take x as 1, it would be a lot simpler, and if im wrong pls tell me why