If α and β are the roots of the equation x2 + x – 1 = 0, then what is the equation whose roots are α5 and β5?
Show answer & explanation
Correct answer: D
Explanation
The correct answer is option 4 i.e. x2 + 11x – 1 = 0
x2 + x – 1 = 0 ; [α + β = -1] ; [αβ = -1]
When roots are α5, β5
Sum of roots = α5 + β5
= (α2 + β2)(α3 + β3) - α2β2(α + β)
= [(α + β)2 - 2αβ][(α + β) [(α + β)2 - 3αβ)]] - α2β2(α + β)
= [(-1)2 - 2(-1)][(-1)[(-1)2 - 3(-1)]] - (-1)2 (-1) = -11 .....(1)
Product of roots = α5β5
=(αβ)5 = (-1)5 = -1
Equation
x2 - (-11)x - 1 = 0
x2 + 11x - 1 = 0