Introduction: When we have the zeroes of a polynomial, we can use them to find the factors of that polynomial. In this case, we are given the zeroes alpha and beta for a quadratic polynomial. We need to find a new quadratic polynomial whose zeroes are 1/alpha and 1/beta.

Using the Zeroes to Find the Factors: We know that if alpha and beta are zeroes of a quadratic polynomial ax2 bx c, then we can write that polynomial in factored form as follows:

ax2 bx c = a(x - alpha)(x - beta)

Reciprocals of Zeroes: We want to find a new quadratic polynomial whose zeroes are 1/alpha and 1/beta. We know that the reciprocals of the zeroes of a polynomial are the zeroes of its reciprocal. Therefore, the zeroes of the reciprocal of ax2 bx c will be 1/alpha and 1/beta.

Finding the Reciprocal of the Polynomial: To find the reciprocal of ax2 bx c, we simply need to divide 1 by each of the terms in the polynomial. This gives us:

1/ax2 bx c = 1/a(x - alpha)(x - beta)

New Polynomial with Reciprocal of Zeroes: Now that we have the reciprocal of the polynomial, we can use it to find a new quadratic polynomial with zeroes 1/alpha and 1/beta. We just need to take the reciprocal of the factored form of the polynomial and simplify. This gives us:

1/ [a(x - alpha)(x - beta)] = (1/a) * (1/(x - alpha)) * (1/(x - beta))

Multiplying the denominators, we get:

1/ [a(x - alpha)(x - beta)] = (1/a) * (1/[(x - alpha) * (x - beta)])

Simplifying, we get:

1/ [a(x - alpha)(x - beta)] = (1/[alpha * beta]) * (1/[(1/alpha) - x] * (1/[(1/beta) - x]))

Therefore, the new quadratic polynomial whose zeroes are 1/alpha and 1/beta is:

(1/[alpha * beta]) * (x - (1/alpha)) * (x - (1/beta))

Conclusion: We have found the new quadratic polynomial whose zeroes are 1/alpha and 1/beta. We used the factored form of the original polynomial, the reciprocal of the original polynomial, and the fact that the reciprocals of the zeroes of a polynomial are the zeroes of its reciprocal.