If α, β and γ are the zeros of the polynomial f(x) = ax³ + bx² + cx + d, find the value of 1/α + 1/β + 1/γ

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 1/α + 1/β + 1/γ.

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: Write the Required Expression

1/α + 1/β + 1/γ

= (βγ + γα + αβ) / (αβγ)

Step 2: Substitute the Values

= (c/a) ÷ (−d/a)

= −c/d

Final Answer

1/α + 1/β + 1/γ = −c/d

Conclusion

Hence, if α, β and γ are the zeros of the polynomial f(x) = ax³ + bx² + cx + d, then the value of 1/α + 1/β + 1/γ is −c/d.