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यदि बहुपद f (x) = x3 – 3px2 + qx – r के मूल समान्तर श्रेणी में हैं तब p,q और r के बीच में सम्बन्ध ज्ञात कीजिए।

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यदि f (x) = x3 – 3px2 + qx – 7

∵ मूल समान्तर श्रेणी में हैं।

अतः माना α = a – d, β = a, γ = a + d

∴ α + β + γ = \(\frac{-b}{a}\)

⇒ a – d + a + a + d = \(\frac{3p}{1}\)

⇒ 3a = 3p

⇒ a = P

तथा α·β + β·γ + γ·α = \(\frac{c}{a}\)

(a – d) × a + a × (a + d) + (a + d) (a – d) = \(\frac{q}{a}\)

a = p रखने पर,

(p – d) × p + px (p + d) + (p + d) (p – d) = q

p2 – pd + p2 + pd + p2 – d2 = q

3p2 – d2 = q

3p2 – q = d2

या d2 = 3p2 – q

अब α·β·γ = \(\frac{-d}{a}\)

(a – d) × a × (a + d) = \(\frac{-(-r)}{1}\)

a × (a2 – d2) = r

a = p तथा d2 = 3p2 – q रखने पर,

p × [p2 – (3p2 – q)] = r

p(p2 – 3p + q) = r

p(-2p2 + q) = r

-2p3 + pq = r

Pq – r = 2p3

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