## Expanding the Expression (x+1)(x^6-x^5+x^4-x^3+x^2-x+1)

This expression represents the product of a binomial and a polynomial. We can expand it by using the distributive property, which means multiplying each term in the binomial with each term in the polynomial.

Let's break down the process:

**Step 1: Distribute (x + 1) to the polynomial**

(x + 1)(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1) = x(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1) + 1(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)

**Step 2: Multiply each term**

- x * x^6 = x^7
- x * -x^5 = -x^6
- x * x^4 = x^5
- x * -x^3 = -x^4
- x * x^2 = x^3
- x * -x = -x^2
- x * 1 = x
- 1 * x^6 = x^6
- 1 * -x^5 = -x^5
- 1 * x^4 = x^4
- 1 * -x^3 = -x^3
- 1 * x^2 = x^2
- 1 * -x = -x
- 1 * 1 = 1

**Step 3: Combine like terms**

x^7 - x^6 + x^5 - x^4 + x^3 - x^2 + x + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 = **x^7 + 1**

**Therefore, the simplified expression of (x+1)(x^6-x^5+x^4-x^3+x^2-x+1) is x^7 + 1.**

**Note:** This is a special case of the sum of powers formula, where the polynomial is a geometric series. In general, the formula for the sum of a geometric series is:

1 + r + r^2 + ... + r^n = (1 - r^(n+1))/(1 - r)

In this case, r = -x, and n = 6.