If α, β be the roots of the quadratic equation x2 + x + 1 = 0 then the equation whose roots are α19, β7 is

Question

If α, β be the roots of the quadratic equation x2 + x + 1 = 0 then the equation whose roots are α19, β7 is (A) x2 - x + 1 = 0 (B) x2 - x - 1 = 0 (C) x2 + x - 1 = 0 (D) x2 + x + 1 = 0

Tardigrade Admin · Accepted Answer

The roots of x2+x+1=0 are ω, ω2 Let α=ω, β=ω2 Now, α19=ω19=ω18+1 =ω18 ω=ω and β7=(ω2)7 =ω12 ω2=ω2 which are the same roots. ∴ Equation x2+x+1=0

Q. If $α, β$ be the roots of the quadratic equation $x^{2} + x + 1 = 0$ then the equation whose roots are $α^{19}, β^{7}$ is

$x^{2} - x + 1 = 0$

$x^{2} - x - 1 = 0$

$x^{2} + x - 1 = 0$

$x^{2} + x + 1 = 0$

The roots of $x^{2} + x + 1 = 0$
are $ω, ω^{2}$
Let $α = ω, β = ω^{2}$
Now, $α^{19} = ω^{19} = ω^{18 + 1} = ω^{18} ω = ω$
and $β^{7} = (ω^{2})^{7} = ω^{12} ω^{2} = ω^{2}$
which are the same roots.
$∴$ Equation $x^{2} + x + 1 = 0$

Q. If α,β be the roots of the quadratic equation x2+x+1=0 then the equation whose roots are α19,β7 is

x2−x+1=0

x2−x−1=0

x2+x−1=0

x2+x+1=0