If α, β and γ are the zeros of the polynomial f(x) = ax³ + bx² + cx + d, find the value of α² + β² + γ²

Video Explanation

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Given

f(x) = ax³ + bx² + cx + d

α, β and γ are the zeros of the polynomial.

To Find

The value of α² + β² + γ².

Solution

For a cubic polynomial:

ax³ + bx² + cx + d

The relationships between zeros and coefficients are:

α + β + γ = −b/a

αβ + βγ + γα = c/a

αβγ = −d/a

Step 1: Use the Identity

α² + β² + γ² = (α + β + γ)² − 2(αβ + βγ + γα)

Step 2: Substitute the Values

= (−b/a)² − 2(c/a)

= b²/a² − 2c/a

Step 3: Write in Simplified Form

α² + β² + γ² = (b² − 2ac) / a²

Final Answer

α² + β² + γ² = (b² − 2ac) / a²

Conclusion

Hence, if α, β and γ are the zeros of the polynomial f(x) = ax³ + bx² + cx + d, then the value of α² + β² + γ² is (b² − 2ac) / a².