Divide (x^5+1)/(x+1)

Problem

(x5+1)/(x+1)

Solution

1. Identify the expression as a division of a sum of two fifth powers, which follows the pattern for an+bn where n is odd.

2. Apply the formula for the sum of powers, which states that xn+1=(x+1)*(x(n−1)−x(n−2)+x(n−3)−⋯+1) for odd integers n

3. Expand the quotient by alternating signs and decreasing the power of x starting from n−1=4

4. Simplify the resulting polynomial expression.

Final Answer

(x5+1)/(x+1)=x4−x3+x2−x+1

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