Question

Is x+1 a factor of x^65+2x^58-3x^35+x^6-4? Explain your answer or show work.

Answer 1

By the polynomial remainder theorem,
`x+1` is a factor of
`x^(65)+2x^(58)-3x^(35)+x^6-4` if the value of the latter polynomial is zero when
`x=-1`.

You have


`(-1)^(65)+2(-1)^(58)-3(-1)^(35)+(-1)^6-4=-1+2+3+1-4=1\\eq0`

so
`x+1` is not a factor.