Condition for Zeroes of a Cubic Polynomial to be in A.P.

Video Explanation

Question

If the zeroes of the polynomial

\[ f(x) = x^3 – 3px^2 + qx – r \]

are in arithmetic progression, find the required condition.

Solution

Step 1: Assume the Zeroes in A.P.

Let the three zeroes be

\[ p-d,\; p,\; p+d \]

where \(p\) is the middle term and \(d\) is the common difference.

Step 2: Use Relations Between Zeroes and Coefficients

For the cubic polynomial \[ x^3 – 3px^2 + qx – r, \]

we have:

\[ \alpha + \beta + \gamma = 3p \]

\[ \alpha\beta + \beta\gamma + \gamma\alpha = q \]

\[ \alpha\beta\gamma = r \]

Step 3: Use A.P. Zeroes

Sum of zeroes:

\[ (p-d) + p + (p+d) = 3p \]

This satisfies the first relation.

Sum of products of zeroes taken two at a time:

\[ (p-d)p + p(p+d) + (p-d)(p+d) \]

\[ = 3p^2 – d^2 \]

So,

\[ 3p^2 – d^2 = q \Rightarrow d^2 = 3p^2 – q \]

Product of zeroes:

\[ (p-d)p(p+d) = p(p^2 – d^2) \]

Substitute \(d^2 = 3p^2 – q\):

\[ p\big(p^2 – (3p^2 – q)\big) = r \]

\[ p(q – 2p^2) = r \]

Step 4: Obtain the Required Condition

\[ pq – 2p^3 = r \]

or,

\[ \boxed{2p^3 – pq + r = 0} \]

Conclusion

The required condition for the zeroes of the polynomial \[ x^3 – 3px^2 + qx – r \] to be in arithmetic progression is:

\[ \boxed{2p^3 – pq + r = 0} \]