Question

4. If a. ß are the zeroes of the polynomial x² + 8x + 6, Form the polynomial whose zeroes are 1/alpha^2 and 1/beta^2​

Answer 1

Q -If α and β are the zeros of the polynomial x2+8x+6 , how do you form the polynomial whose zeros are 1+βα and 1+αβ ?

This question is regarding Sum and Products of roots of a Quadratic equation.

First, we need to know the formulae for Sum and Product of roots of a Quadratic Equation.

Consider an equation, ax2+bx+c=0 , it’s roots be alpha and beta .

So, Sum of Roots, alpha+beta=−b/a

Product of Roots, alpha.beta=c/a

So, here, we have an equation, x2+8x+6 , whose roots are alpha and beta .

We need to find a polynomial with roots 1+(beta/alpha) and 1+(alpha/beta)

We know that, 1+(beta/alpha)=(alpha+beta)/alpha

Also, 1+(alpha/beta)=(alpha+beta)/beta

So, here, alpha+beta=−8

So, 1+(beta/alpha)=−8/alpha and 1+(alpha/beta)=−8/beta

We can reduce the question as, find the equation whose roots are −8/alpha and −8/beta .

Let, d=−8/alpha,−8/beta

So,

alpha=−8/y and beta=−8/y

Put values in original equation, we get,

(64/y2)−(64/y)+6=0

Simplifying this, we get the equation as,

3y2−32y+32=0