﻿ Let a, b, p, q Ïµ Q and suppose that f (x) = x2 + ax + b = 0 and g (x) = x3+ px + q = 0 have a common irrational root, then : Kaysons Education

# Let a, b, p, q Ïµ Q and Suppose That f (x) = x2 + ax + b = 0 And g (x) = x3+ px + q = 0 Have A Common Irrational Root, Then

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## Question

### Solution

Correct option is

f (x) divides g (x)

Let α Ïµ R – Q be a common root of f (x) = 0 and g (x) = 0. Then α2 = –aα – b. substituting this in a3 + pα + q = 0, we get

(a2 – b + p) α + ab + q = 0

As α is irrational and abpq Ïµ Qp = b – a2q = – ab. This gives, g(x) = (x – af (x).

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