Question

If alpha and beta are the zeroes of x^2+ x +5then find value of alpha^4 beta^4

Answer 1

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.)