Question

find the polynomial whose zeroes.are.reciprocals of the zeroes of.the polynomial 2x2 + 3x -6

Answer 1

Let p& q be zeoes of 2x2 + 3x -6
p+q=-3/2, p*q=-3
for the new polynomial,
sum of zeroes= 1/p+1/q
=(p+q)/pq
=-3/2/-3=1/2
product of zeroes=1/p*1/q=1/pq=-1/3
so the new polynomial is
x^2-1/2x-1/3 or 6x^2-3x-2

Answer 2

Given us the polynomial, 2x²+3x-6. so, let the zeroes of the mentioned polynomial are α and β. Hence, α+β = -3/2 and αβ = -6/2 = -3. We know the reciprocals of zeroes are 1/α and 1/β. Now, the sum of the zeroes are 1/α+1/β = α+β/αβ = -3/2×1/-3 = 1/2, again, the product of the zeroes are 1/α×1/β = 1/αβ = 1/-3. So, the polynomial is,
                       x²-(1/α+1/β)x+1/α×1/β= 0 
                   ⇒ x²-(1/2)x+1/-3= 0
                   ⇒ x²-1/2x-1/3= 0
                   ⇒ 6x²-3x-2/6= 0
                   ⇒ 6x²-3x-2= 0
The answer is 6x²-3x-2= 0