If the zeros of the polynomial f(x) = x³ − 3px² + qx − r are in A.P., find the required condition

Video Explanation

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Given

f(x) = x³ − 3px² + qx − r

The zeros of the polynomial are in Arithmetic Progression (A.P.).

To Find

The condition among p, q and r.

Solution

Step 1: General Condition for Zeros in A.P.

For a cubic polynomial:

x³ + ax² + bx + c

If its zeros are in A.P., then the condition is:

2a³ − 9ab + 27c = 0

Step 2: Compare with the Given Polynomial

Given polynomial:

x³ − 3px² + qx − r

Comparing with x³ + ax² + bx + c, we get:

a = −3p
b = q
c = −r

Step 3: Substitute in the Condition

2a³ − 9ab + 27c = 0

⇒ 2(−3p)³ − 9(−3p)(q) + 27(−r) = 0

⇒ 2(−27p³) + 27pq − 27r = 0

⇒ −54p³ + 27pq − 27r = 0

Dividing throughout by 27:

−2p³ + pq − r = 0

or

2p³ − pq + r = 0

Final Answer

The required condition is:

2p³ − pq + r = 0

Conclusion

Hence, if the zeros of the polynomial f(x) = x³ − 3px² + qx − r are in Arithmetic Progression, then the condition among p, q and r is 2p³ − pq + r = 0.