If alpha amd beta are zeroes of polynomial
x^2+8x-6 form the polynomial whose zeroes are 1 /2alpha and 1/2beta.
f(x) = x^2+8x-6
if the roots are 1/2 as much, then they are the roots of
f(2x) = 4x^2+16x-6
I'll let you check with the quadratic formula.
To find the polynomial whose zeroes are 1/2alpha and 1/2beta, we can make use of the fact that if alpha is a zero of a polynomial, then x - alpha is a factor of that polynomial.
Given the zeroes alpha and beta of the polynomial x^2 + 8x - 6, we can write the factors as (x - alpha) and (x - beta).
To find the polynomial whose zeroes are 1/2alpha and 1/2beta, we need to consider the reciprocals of the above factors. That is, the factors for the new polynomial will be (x - 1/(2alpha)) and (x - 1/(2beta)).
To simplify, we can express the reciprocals as fractions, making the factors: (2x - 1/alpha) and (2x - 1/beta).
Now, multiplying these factors together, we get the polynomial:
P(x) = (2x - 1/alpha) * (2x - 1/beta)
= (2x - 1/alpha) * (2x - 1/beta)
= (2x - 1) / alpha * (2x - 1) / beta
Expanding this expression, we get:
P(x) = (4x^2 - 2x/alpha - 2x/beta + 1/alpha*beta)
Therefore, the polynomial whose zeroes are 1/2alpha and 1/2beta is:
P(x) = 4x^2 - 2x/alpha - 2x/beta + 1/alpha*beta