Answer:
Answer: None
Step-by-step explanation:
Polynomial Remainder Theorem
The polynomial remainder theorem establishes that the remainder of the division of a polynomial f(x) by (x-k) is equal to f(k), that is, substituting x for k.
We are given the polynomial:
[tex]f(x)= x^4+x^3-x^2+x-1[/tex]
And it's required to know if (x+1) and/or (x-1) are factors of the polynomial.
We apply the theorem for both cases.
For (x+1), we substitute x=-1 into the polynomial:
[tex]f(-1)= (-1)^4+(-1)^3-(-1)^2+(-1)-1[/tex]
[tex]f(-1)= 1-1-1-1-1=-3[/tex]
Since the remainder is not zero, x+1 is not a factor of the given polynomial.
For (x-1), we substitute x=1 into the polynomial:
[tex]f(-1)= (1)^4+(1)^3-(1)^2+(1)-1[/tex]
[tex]f(-1)= 1+1-1+1-1=1[/tex]
Since the remainder is not zero, x-1 is not a factor of the given polynomial.
Answer: None
