If alpha and beta are zeroes of polynomial x^2+5x+5,evaluate the following. ALPHA SQUARE+BETA SQUARE ALPHA CUBE+BETA CUBE alpha^-2+beta^-2
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α,β are the zeros of the polynomial x²+5x+5.
∴, α+β=-5/1=-5 and αβ=5/1=5
∴, α²+β²
=(α+β)²-2αβ
=(-5)²-2×5
=25-10
=15
α³+β³
=(α+β)³-3αβ(α+β)
=(-5)³-3×5×(-5)
=-125+75
=-50
α⁻²+β⁻²
=1/α²+1/β²
=(β²+α²)/α²β²
={(α+β)²-2αβ}/(αβ)²
={(-5)²-2×5}/(5)²
=(25-10)/25
=15/25
=3/5
∴, α+β=-5/1=-5 and αβ=5/1=5
∴, α²+β²
=(α+β)²-2αβ
=(-5)²-2×5
=25-10
=15
α³+β³
=(α+β)³-3αβ(α+β)
=(-5)³-3×5×(-5)
=-125+75
=-50
α⁻²+β⁻²
=1/α²+1/β²
=(β²+α²)/α²β²
={(α+β)²-2αβ}/(αβ)²
={(-5)²-2×5}/(5)²
=(25-10)/25
=15/25
=3/5
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