Use the Remainder Theorem to find the remainder when f(x) is divided by x - c. f(x) = x^{4} + 8x^3+ 11x^2 ; x + 1x+1

Answer 1

The Remainder Theorem says that the remainder, R, when a polynomial, p(x), is divided by the linear factor (x - c), is = p(c).

This is an easy theorem to prove: p(x) = Q(x)(x - c) + R, where Q(x) is the quotient polynomial. Then substitute c in for x. This theorem also leads directly to the Factor Theorem, which says (x - c) is a factor of p(x) iff p(c) = 0.

The Remainder Theorem can sometimes be used, as in this case, to find a remainder by direct substitution. It can also be used to find the value of a function by doing a synthetic division. In all cases, these situations are frequently of most interest when f(c) = R = 0, because it means we have a zero of the polynomial.

So: R = f(-1) = 1 - 8 + 11 = 4.

Answer 2

By Remainder Theore this remainder r = f(- 1) = 1 - 8 + 11 = 4