polynomial

2009-07-19 11:38 pm

β±

Let f(x) be a polynomial with integer coefficients.
Suppose β an integer root of f(x) and p is a prime number.

Show that the only possible integer roots of the polynomial [f(x) + p] are :

β ± p and β ±1

回答 (1)

myisland8132

2009-07-20 2:51 am

✔ 最佳答案

As β is a integer root of f(x), therefore:

f(x) = (x-β) g(x) ....... (1)

where g(x) is a polynomial with integer coefficients

Let b be the integer root of f(x) + p, then:

f(x) + p = (x-b) h(x) where h(x) is a polynomial with integer coefficients.

f(b) + p = 0

or f(b) = -p ............(2)

Put x= b into equation (1):

f(b) = (b-β) g(b) ......... (3)

From (2) and (3):

(b-β) g(b) = -p

b = -p/g(b) + β

As p is a prime number, g(b) can only be p or 1.

If g(b) = p, b = β1

If g(b) = 1,b = βp

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