Question

The remainder when

(x ^n )+n divided by x-1 is :​

Answer 1

Step-by-step explanation:

To divide f(X)=f(X)=X−1X−1 into g(X)=Xn−1g(X)=Xn−1, use the division algorithm. Recall that the division algorithm states that there exists polynomials q(X),r(X)q(X),r(X) such that g(X)=q(X)f(X)+r(X)g(X)=q(X)f(X)+r(X). The polynomial r(X)r(X) is the remainder. This yields:

Xn−1=(X−1)(Xn−1+Xn−2+⋯+X2+X+1)+0Xn−1=(X−1)(Xn−1+Xn−2+⋯+X2+X+1)+0

So, according to the algorithm, q(X)=Xn−1+Xn−2+⋯+X2+X+1q(X)=Xn−1+Xn−2+⋯+X2+X+1 and r(X)=0r(X)=0. So, the remainder is 0, which is equivalent to noticing that f(X)f(X) divides into g(X)g(X).

To see the equation which is yielded, simply factorise g(X)g(X), but ensuring that X−1X−1 is one of the factors. Starting with the observation Xn=XXn−1Xn=XXn−1 then this means we gain a −Xn−1