If α and β are the zeros of the quadratic polynomial f(t) = t² − 4t + 3, find the value of (α⁴β³ + α³β⁴)

Video Explanation

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Solution

Given polynomial:

f(t) = t² − 4t + 3

Step 1: Find α + β and αβ

Comparing f(t) = t² − 4t + 3 with at² + bt + c:

a = 1,   b = −4,   c = 3

α + β = −b/a = 4

αβ = c/a = 3

Step 2: Find the Required Value

α⁴β³ + α³β⁴

= α³β³(α + β)

= (αβ)³(α + β)

= 3³ × 4

= 27 × 4

= 108

Final Answer

The required value is 108.

Conclusion

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