If α and β are the zeros of the quadratic polynomial f(x) = x² − 2x + 3, find the polynomial whose roots are (α − 1)/(α + 1) and (β − 1)/(β + 1)

Video Explanation

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Solution

Given polynomial:

f(x) = x² − 2x + 3

Step 1: Find α + β and αβ

Comparing f(x) = x² − 2x + 3 with ax² + bx + c:

a = 1,   b = −2,   c = 3

α + β = −b/a = 2

αβ = c/a = 3

Step 2: Find the Sum of the New Roots

Sum of new roots:

(α − 1)/(α + 1) + (β − 1)/(β + 1)

= [(α − 1)(β + 1) + (β − 1)(α + 1)] / [(α + 1)(β + 1)]

= [αβ + α − β − 1 + αβ + β − α − 1] / [αβ + α + β + 1]

= (2αβ − 2) / (αβ + α + β + 1)

= (2 × 3 − 2) / (3 + 2 + 1)

= 4/6 = 2/3

Step 3: Find the Product of the New Roots

Product of new roots:

[(α − 1)/(α + 1)] × [(β − 1)/(β + 1)]

= (αβ − α − β + 1) / (αβ + α + β + 1)

= (3 − 2 + 1) / 6

= 2/6 = 1/3

Step 4: Form the Required Polynomial

The quadratic polynomial whose roots are (α − 1)/(α + 1) and (β − 1)/(β + 1) is:

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

= x² − (2/3)x + 1/3

Multiplying throughout by 3:

3x² − 2x + 1

Final Answer

The required polynomial is 3x² − 2x + 1.

Conclusion

Thus, the quadratic polynomial whose roots are (α − 1)/(α + 1) and (β − 1)/(β + 1) is 3x² − 2x + 1.