(x%5E2%2Bx%2B1).jpeg "(x-1)(x^2+x+1)")
Factoring and Expanding (x-1)(x^2+x+1)
This expression involves the multiplication of two factors: (x-1) and (x^2+x+1). We can explore this expression through factoring and expanding.
Factoring the Expression
The expression (x-1)(x^2+x+1) is already factored. It's presented as the product of two factors.
Expanding the Expression
To expand the expression, we need to apply the distributive property:
1. Distribute the first term (x):
x * (x^2 + x + 1) = x^3 + x^2 + x
2. Distribute the second term (-1):
-1 * (x^2 + x + 1) = -x^2 - x - 1
3. Combine the results:
x^3 + x^2 + x - x^2 - x - 1
4. Simplify by combining like terms:
x^3 - 1
Therefore, the expanded form of (x-1)(x^2+x+1) is x^3 - 1.
Key Observations
- The expression (x^2+x+1) is a special trinomial known as the sum of cubes pattern: a^3 + b^3 = (a+b)(a^2 - ab + b^2). In this case, a = x and b = 1.
- The expanded form (x^3 - 1) is also a difference of cubes: a^3 - b^3 = (a-b)(a^2 + ab + b^2).
Understanding these patterns can be helpful in factoring and expanding similar expressions.