Polynomial from Given Zeros

Video Explanation

Question

If \( \alpha \) and \( \beta \) are the zeros of the polynomial

\[ f(x) = x^2 + px + q, \]

form a polynomial whose zeros are

\[ (\alpha + \beta)^2 \quad \text{and} \quad (\alpha – \beta)^2. \]

Solution

Step 1: Write Relations Between Zeros and Coefficients

For \(x^2 + px + q\),

\[ \alpha + \beta = -p, \quad \alpha\beta = q \]

Step 2: Find the New Zeros

\[ (\alpha + \beta)^2 = (-p)^2 = p^2 \]

\[ (\alpha – \beta)^2 = (\alpha + \beta)^2 – 4\alpha\beta = p^2 – 4q \]

Step 3: Find Sum and Product of the New Zeros

Sum:

\[ S = p^2 + (p^2 – 4q) = 2p^2 – 4q \]

Product:

\[ P = p^2(p^2 – 4q) \]

Step 4: Form the Required Polynomial

A polynomial with sum \(S\) and product \(P\) of zeros is:

\[ x^2 – Sx + P \]

\[ = x^2 – (2p^2 – 4q)x + p^2(p^2 – 4q) \]

Conclusion

The required polynomial is:

\[ \boxed{x^2 – (2p^2 – 4q)x + p^2(p^2 – 4q)} \]

\[ \therefore \quad \text{This is the required polynomial.} \]