If α, β and γ are the zeros of the polynomial f(x) = x³ − px² + qx − r, find the value of 1/αβ + 1/βγ + 1/γα

Video Explanation

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Given

f(x) = x³ − px² + qx − r

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

To Find

The value of 1/αβ + 1/βγ + 1/γα.

Solution

For the cubic polynomial:

x³ − px² + qx − r

The relationships between zeros and coefficients are:

α + β + γ = p

αβ + βγ + γα = q

αβγ = r

Step 1: Write the Required Expression

1/αβ + 1/βγ + 1/γα

= (α + β + γ) / (αβγ)

Step 2: Substitute the Values

= p / r

Final Answer

1/αβ + 1/βγ + 1/γα = p/r

Conclusion

Hence, if α, β and γ are the zeros of the polynomial f(x) = x³ − px² + qx − r, then the value of 1/αβ + 1/βγ + 1/γα is p/r.