Quadratic Polynomial with zeroes 1/alpha and 1/beta

To find a quadratic polynomial with zeroes 1/alpha and 1/beta, we need to use the relationship between the zeroes and coefficients of a quadratic polynomial.

1. Understanding the relationship between zeroes and coefficients
In a quadratic polynomial of the form ax^2 + bx + c, the sum of the zeroes is given by -b/a, and the product of the zeroes is given by c/a.

For the given polynomial x^2 - 3x + 7, the sum of the zeroes is -(-3)/1 = 3, and the product of the zeroes is 7/1 = 7.

2. Finding the sum and product of the reciprocals
To find the sum and product of 1/alpha and 1/beta, we can use the reciprocal property.

The sum of the reciprocals is 1/alpha + 1/beta = (alpha + beta)/(alpha * beta), and the product of the reciprocals is 1/alpha * 1/beta = 1/(alpha * beta).

3. Finding the sum of the reciprocals
Using the sum of the zeroes formula, we know that alpha + beta = -b/a = 3.

Therefore, the sum of the reciprocals is 1/alpha + 1/beta = (alpha + beta)/(alpha * beta) = 3/(alpha * beta).

4. Finding the product of the reciprocals
Using the product of the zeroes formula, we know that alpha * beta = c/a = 7/1 = 7.

Therefore, the product of the reciprocals is 1/alpha * 1/beta = 1/(alpha * beta) = 1/7.

5. Writing the quadratic polynomial
Now, we have the sum of the reciprocals as 3/(alpha * beta) and the product of the reciprocals as 1/7.

So, the quadratic polynomial with zeroes 1/alpha and 1/beta is (x - 3/(alpha * beta))(x - 1/7).

Simplifying this expression, we get:

(x - 3/(alpha * beta))(x - 1/7) = (7x - 3)/(7 * alpha * beta)

Therefore, the quadratic polynomial with zeroes 1/alpha and 1/beta is (7x - 3)/(7 * alpha * beta).

Summary:
To find a quadratic polynomial with zeroes 1/alpha and 1/beta, we used the relationship between the zeroes and coefficients of a quadratic polynomial. We found that the quadratic polynomial is given by (7x - 3)/(7 * alpha * beta).