If α and β are the zeros of the quadratic polynomial p(s) = 3s² − 6s + 4, find the value of α/β + β/α + 2(1/α + 1/β) + 3αβ

Video Explanation

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Solution

Given polynomial:

p(s) = 3s² − 6s + 4

Step 1: Find α + β and αβ

Comparing p(s) = 3s² − 6s + 4 with as² + bs + c:

a = 3,   b = −6,   c = 4

α + β = −b/a = 6/3 = 2

αβ = c/a = 4/3

Step 2: Find Each Required Term

α/β + β/α

= (α² + β²)/αβ

= {(α + β)² − 2αβ}/αβ

= {2² − 2(4/3)}/(4/3)

= (4 − 8/3)/(4/3)

= 1

1/α + 1/β = (α + β)/αβ

= 2 ÷ (4/3)

= 3/2

2(1/α + 1/β) = 3

3αβ = 3 × 4/3 = 4

Step 3: Find the Required Value

α/β + β/α + 2(1/α + 1/β) + 3αβ

= 1 + 3 + 4

= 8

Final Answer

The required value is 8.

Conclusion

Thus, using the relationship between zeros and coefficients of the quadratic polynomial, the value of the given expression is correctly obtained.