Let f ( x ) = x n + 5 x n − 1 + 3 where n ≥ 1 is an integer. Prove that f ( x ) can't be expressed as the product of two polynomials each of which has all its coefficients integers and degree ≥ 1.

Question

Let   f  (  x  )  =      x    n    +  5      x          n      −      1        +  3 where   n  ≥  1 is an integer. Prove that   f  (  x  ) can&#039;t be expressed as the product of two polynomials each of which has all its coefficients integers and degree   ≥  1.

h2a2l1i2morz · Accepted Answer

The notation   [      x    t    ]  P  (  x  ) denotes the coefficient of       x    t   in polynomial   P  (  x  )Suppose that   f  (  x  )  =  g  (  x  )  h  (  x  ), where   g  (  x  )  ,  h  (  x  ) are monic polynomials,   g  (  x  )  ,  h  (  x  )  ∈      Z    [  x  ] and   deg  ⁡  g  &amp;gt;  0,   deg  ⁡  h  &amp;gt;  0,   deg  ⁡  g  +  deg  ⁡  h  =  n. Let   g  (  x  )  =      ∑    k        α    k        x    k  ,   h  (  x  )  =      ∑    k        β    k        x    k  , where       α    k    ,      β    k   is integer whenever   k  ≥  0First, we have       α    0        β    0    =  3, so       |        α    0        |    =  3 and       |        β    0        |    =  1 or       |        α    0        |    =  1 and       |        β    0        |    =  3. WLOG, suppose that       |        α    0        |    =  3 and       |        β    0        |    =  1. Let m is the smallest positive integer such that   3  ∤      α    m   (such m exists because g(x) is monic). Now we have  [      z    m    ]  f  (  x  )  =      α    0        β    m    +      α    1        β          m      −      1        +  ⋯  +      α    m        β    0    ≡      α    m        β    0    ≢  0    (  mod    3  )So   m  ≥  n  −  1, and   deg  ⁡  g  ≥  n  −  1, thus   deg  ⁡  h  ≤  1, hence   deg  ⁡  h  =  1, and   h  (  x  )  =  x  +      β    0  , where       β    0    =  ±  1, so   h  (      β    0    )  =  0, and   f  (      β    0    )  =  0, but it&#039;s obvious that   f  (  ±  1  )  ≠  0, Q.E.D.

Let f(x)=x^n+5x^(n−1)+3 where n≥1 is an integer. Prove that f(x) can't be expressed as the product of two polynomials each of which has all its coefficients integers and degree >=1.

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