Expanding the Expression (x+1)(x^2+2x+1)
This article will explore the process of expanding the given expression: (x+1)(x^2+2x+1). We will use the distributive property and simplify the resulting expression.
Understanding the Distributive Property
The distributive property states that multiplying a sum by a number is the same as multiplying each addend separately and then adding the products. In mathematical terms:
a(b+c) = ab + ac
Expanding the Expression

Apply the distributive property: We treat (x+1) as a single term and multiply it by each term within the second set of parentheses:
(x+1)(x^2+2x+1) = x(x^2+2x+1) + 1(x^2+2x+1)

Distribute again: Now we distribute x and 1 to each term inside the parentheses:
x(x^2+2x+1) + 1(x^2+2x+1) = x^3 + 2x^2 + x + x^2 + 2x + 1

Combine like terms: Finally, we combine the terms with the same exponents:
x^3 + 2x^2 + x + x^2 + 2x + 1 = x^3 + 3x^2 + 3x + 1
Conclusion
Therefore, the expanded form of the expression (x+1)(x^2+2x+1) is x^3 + 3x^2 + 3x + 1. This process demonstrates the use of the distributive property to expand expressions and combine like terms.