If α and β are the zeros of the polynomial f(x) = x² + px + q, find the polynomial whose zeros are (α + β)² and (α − β)²

Video Explanation

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Solution

Given polynomial:

f(x) = x² + px + q

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

Step 1: Find α + β and αβ

Comparing f(x) = x² + px + q with ax² + bx + c:

a = 1,   b = p,   c = q

α + β = −b/a = −p

αβ = c/a = q

Step 2: Find the New Zeros

First zero:

(α + β)² = (−p)² = p²

Second zero:

(α − β)² = (α + β)² − 4αβ

= p² − 4q

Step 3: Find Sum and Product of the New Zeros

Sum of new zeros:

p² + (p² − 4q) = 2p² − 4q

Product of new zeros:

p²(p² − 4q)

Step 4: Form the Required Polynomial

The required polynomial is:

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

= x² − (2p² − 4q)x + p²(p² − 4q)

Final Answer

The required polynomial is x² − (2p² − 4q)x + p²(p² − 4q).

Conclusion

Thus, the polynomial whose zeros are (α + β)² and (α − β)² is x² − (2p² − 4q)x + p²(p² − 4q).