If α and β are the zeros of the polynomial f(x) = x² + px + q, find the polynomial whose zeros are 1/α and 1/β

Video Explanation

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Given

f(x) = x² + px + q

Let α and β be the zeros of the given polynomial.

To Find

A polynomial whose zeros are 1/α and 1/β.

Solution

For the quadratic polynomial:

x² + px + q

Sum of zeros:

α + β = −p

Product of zeros:

αβ = q

Step 1: Find Sum and Product of New Zeros

New zeros are:

1/α and 1/β

Sum of new zeros:

1/α + 1/β = (α + β)/(αβ)

= (−p)/q

Product of new zeros:

(1/α)(1/β) = 1/(αβ)

= 1/q

Step 2: Form the Required Polynomial

Required polynomial is:

x² − (sum of zeros)x + (product of zeros)

= x² − (−p/q)x + 1/q

= x² + (p/q)x + 1/q

Multiplying throughout by q:

qx² + px + 1

Final Answer

The required polynomial whose zeros are 1/α and 1/β is:

qx² + px + 1

Conclusion

Hence, if α and β are the zeros of the polynomial x² + px + q, then the polynomial having zeros 1/α and 1/β is qx² + px + 1.