If alpha and beta are the zeroes of x^2+ x +5then find value of alpha^4 beta^4
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Polynomial in variable xx:
p(x)=x2−4p(x)=x2−43–√x+33x+3
Given,
α,βα,β are the zeroes of p(x)p(x).
p(x)p(x) is a quadratic polynomial with a degree 22
From the properties of a quadratic polynomial, we know the relation between quotients (those constants except for variables) and zeroes — α,βα,β .
In p(x)p(x), a=1|b=−43–√|c=3a=1|b=−43|c=3
sum of Zeroes:-
α+β=−ba→α+β=−ba→
−(−43√)1−(−43)143–√43α+β=43–√α+β=43——————— Eq.1
Product of Zeroes :-
αβ=caαβ=ca
3131αβ=3αβ=3—————Eq.2
(α+β)−(αβ)=?(α+β)−(αβ)=?
43–√−343−3
(α+β)−(αβ)=43–√−(α+β)−(αβ)=43−3≈(4×1.732)−33≈(4×1.732)−3
≈6.92−3≈6.92−3
≈3.92≈3.92
∴(α+β)−(αβ)=43–√−3≈3.92∴(α+β)−(αβ)=43−3≈3.92 (approx.)
p(x)=x2−4p(x)=x2−43–√x+33x+3
Given,
α,βα,β are the zeroes of p(x)p(x).
p(x)p(x) is a quadratic polynomial with a degree 22
From the properties of a quadratic polynomial, we know the relation between quotients (those constants except for variables) and zeroes — α,βα,β .
In p(x)p(x), a=1|b=−43–√|c=3a=1|b=−43|c=3
sum of Zeroes:-
α+β=−ba→α+β=−ba→
−(−43√)1−(−43)143–√43α+β=43–√α+β=43——————— Eq.1
Product of Zeroes :-
αβ=caαβ=ca
3131αβ=3αβ=3—————Eq.2
(α+β)−(αβ)=?(α+β)−(αβ)=?
43–√−343−3
(α+β)−(αβ)=43–√−(α+β)−(αβ)=43−3≈(4×1.732)−33≈(4×1.732)−3
≈6.92−3≈6.92−3
≈3.92≈3.92
∴(α+β)−(αβ)=43–√−3≈3.92∴(α+β)−(αβ)=43−3≈3.92 (approx.)
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