If α and β are the zeroes of the quadratic polynomial f(x) = ax² + bx + c, find the value of α⁴ + β⁴

Video Explanation

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Solution

Given polynomial:

f(x) = ax² + bx + c

Let α and β be the zeroes of the given quadratic polynomial.

Step 1: Write the Known Relations

For a quadratic polynomial ax² + bx + c:

α + β = −b/a

αβ = c/a

Step 2: Use the Identity for α⁴ + β⁴

α⁴ + β⁴ = (α² + β²)² − 2α²β²

Also,

α² + β² = (α + β)² − 2αβ

Step 3: Substitute the Values

α² + β² = (−b/a)² − 2(c/a)

= (b² − 2ac)/a²

α²β² = (αβ)² = c²/a²

Step 4: Final Calculation

α⁴ + β⁴

= [(b² − 2ac)/a²]² − 2(c²/a²)

= (b⁴ − 4ab²c + 4a²c²)/a⁴ − 2c²/a²

= (b⁴ − 4ab²c + 2a²c²)/a⁴

Final Answer

The required value is (b⁴ − 4ab²c + 2a²c²) / a⁴.

Conclusion

Thus, using the relationship between zeroes and coefficients of the quadratic polynomial, the value of α⁴ + β⁴ is (b⁴ − 4ab²c + 2a²c²) / a⁴.