(4x%5E2%2B2x%2B1)-7(x%5E3%2B1).jpeg "(2x-1)(4x^2+2x+1)-7(x^3+1)")
Simplifying the Expression: (2x-1)(4x^2+2x+1)-7(x^3+1)
This expression involves the multiplication of polynomials and the subtraction of terms. Let's break it down step-by-step to simplify it.
Expanding the Products
First, we need to expand the products:
- (2x-1)(4x^2+2x+1): This is a special case of the "sum and difference of cubes" pattern.
- (a-b)(a^2+ab+b^2) = a^3 - b^3
- In this case, a = 2x and b = 1.
- So, (2x-1)(4x^2+2x+1) = (2x)^3 - (1)^3 = 8x^3 - 1
- 7(x^3+1): This is a simple distributive property.
- 7(x^3+1) = 7x^3 + 7
Combining Like Terms
Now, let's combine the expanded terms:
(8x^3 - 1) - (7x^3 + 7) = 8x^3 - 1 - 7x^3 - 7
Finally, combine the like terms:
8x^3 - 7x^3 - 1 - 7 = x^3 - 8
Final Simplified Expression
Therefore, the simplified form of the given expression is x^3 - 8.