## Question

Let *a*, *b*, *c*, *d* be four integers such that *ad* is odd and *bc* is even, then *ax*^{3}+ *bx*^{2} + *cx* + *d* = 0 ... (1)

has

### Solution

At least one irrational root

Putting *ax* = *y* (1) can be written as

*y*^{3} + *by*^{2} + *acy* + *a*^{2}*d* = 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 *a*^{2} *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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