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

Jasonbradley

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Jun 16, 2024

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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.