According to the Remainder theorem (see below), the remainder after division any polynomial f(x) by a binomial (x-1)
is equal to the value of the polynomial f(x) at x= 1, i.e. f(1)
In your case,
f(1) = (-8x^25+x^11-x^4+9x) at (x= 1) = -8^1^25 + 1^11 - 1^4 + 9*1 = -8 + 1 - 1 + 9 = 1.
Answered, solved and completed.
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Theorem (the remainder theorem)
1. The remainder of division the polynomial by the binomial is equal to the value of the polynomial.
2. The binomial divides the polynomial if and only if the value of is the root of the polynomial , i.e. .
3. The binomial factors the polynomial if and only if the value of is the root of the polynomial , i.e. .
