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Question

If roots of polynomial ax^3+3bx^2+3cx+d are in A.p. prove that 2b^3-3abc+a^2d=0

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Solution

Let the roots be p,q, and r.
We know that
1. p.q.r = -d/a
2. p+q+r = -3b/a
3. pq+qr+rp=3c/a

Since the roots are in AP, we can assume the roots to p-m, p and p+m (where m is the common difference)
∴ p-m+p+p+m = -3b/a
3p=-3b/a
p=-b/a

(p-m)(p)(p+m)=-d/a
p.(p²-m²)=-d/a
since p=-b/a, we get p²-m²=d/b

Applying the third condition:
p(p-m)+p(p+m)+(p-m)(p+m)=3c/a
2p²+(p²-m²)=3c/a

substituting for p and p²-m²

2b²/a² + d/b = 3c/a

Simplifying this gives:
2b³+a²d=3abc

2b³-3abc+a²d=0


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