﻿ Let a, b, c, d be four integers such that ad is odd and bc is even, then ax3+ bx2 + cx + d = 0       ... (1)       has : Kaysons Education

# Let a, b, c, d be Four Integers Such That ad is Odd And bc is Even, Then ax3+ bx2 + cx + d = 0       ... (1)       Has

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## Question

### Solution

Correct option is

At least one irrational root

Putting ax = y (1) can be written as

y3 + by2 + acy + a2d = 0         ... (2)

If (1) has all rational roots then (2) has all integral roots. If α, β, γ are roots of (2) then α, β, γ are divisors of a2 d and as such must be odd integers. Now

– b = α + β + γ ⇔ b is odd

and               ac = βγ + γα + αβ ⇔ c is odd

⇒                bc is odd. A contradiction.

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