If α and β are the zeros of the quadratic polynomial f(x) = x² − 1, find the quadratic polynomial whose zeros are 2α/β and 2β/α

Video Explanation

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Solution

Given polynomial:

f(x) = x² − 1

Step 1: Find α + β and αβ

Comparing f(x) = x² − 1 with ax² + bx + c:

a = 1,   b = 0,   c = −1

α + β = −b/a = 0

αβ = c/a = −1

Step 2: Find the Sum of the New Zeros

Sum of new zeros:

2α/β + 2β/α

= 2(α² + β²)/αβ

Now,

α² + β² = (α + β)² − 2αβ

= 0 − 2(−1)

= 2

∴ Sum of new zeros

= 2 × 2 / (−1)

= −4

Step 3: Find the Product of the New Zeros

Product of new zeros:

(2α/β)(2β/α)

= 4(αβ)/(αβ)

= 4

Step 4: Form the Required Quadratic Polynomial

The quadratic polynomial whose zeros are 2α/β and 2β/α is:

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

= x² − (−4)x + 4

= x² + 4x + 4

Final Answer

The required quadratic polynomial is x² + 4x + 4.

Conclusion

Thus, the quadratic polynomial whose zeros are 2α/β and 2β/α is x² + 4x + 4.