Quadratic Polynomial from Given Zeros

Video Explanation

Question

If \( \alpha \) and \( \beta \) are the zeros of a quadratic polynomial such that

\[ \alpha + \beta = 24 \quad \text{and} \quad \alpha – \beta = 8, \]

find a quadratic polynomial having \( \alpha \) and \( \beta \) as its zeros.

Solution

Step 1: Find the Values of \( \alpha \) and \( \beta \)

Given:

\[ \alpha + \beta = 24 \quad (1) \]

\[ \alpha – \beta = 8 \quad (2) \]

Adding equations (1) and (2):

\[ 2\alpha = 32 \Rightarrow \alpha = 16 \]

Substitute \( \alpha = 16 \) in equation (1):

\[ 16 + \beta = 24 \Rightarrow \beta = 8 \]

Step 2: Write the Quadratic Polynomial

A quadratic polynomial having zeros \( \alpha \) and \( \beta \) is:

\[ (x – \alpha)(x – \beta) \]

\[ = (x – 16)(x – 8) \]

\[ = x^2 – 24x + 128 \]

Conclusion

The required quadratic polynomial is:

\[ \boxed{x^2 – 24x + 128} \]

\[ \therefore \quad x^2 – 24x + 128 \text{ is the required polynomial.} \]