(4x%5E2%2B2x%2B1)%2B8(x-1)(x%5E2%2Bx%2B1).jpeg "(1-2x)(4x^2+2x+1)+8(x-1)(x^2+x+1)")
Expanding and Simplifying the Expression: (1-2x)(4x^2+2x+1)+8(x-1)(x^2+x+1)
This expression involves expanding two products of binomials and trinomials. Let's break down the process step-by-step.
Step 1: Expanding the first product
We have (1-2x)(4x^2+2x+1). This resembles the pattern of a difference of cubes, where:
- a = 1
- b = 2x
Applying the formula (a-b)(a^2+ab+b^2) = a^3 - b^3, we get:
(1-2x)(4x^2+2x+1) = 1^3 - (2x)^3 = 1 - 8x^3
Step 2: Expanding the second product
Similarly, we have 8(x-1)(x^2+x+1). This also follows the pattern of a difference of cubes, with:
- a = x
- b = 1
Applying the formula, we get:
8(x-1)(x^2+x+1) = 8(x^3 - 1^3) = 8x^3 - 8
Step 3: Combining the expanded terms
Now, let's combine the results from Step 1 and Step 2:
(1-2x)(4x^2+2x+1)+8(x-1)(x^2+x+1) = (1 - 8x^3) + (8x^3 - 8)
Finally, simplifying the expression by combining like terms:
1 - 8x^3 + 8x^3 - 8 = -7
Conclusion
Therefore, the simplified form of the expression (1-2x)(4x^2+2x+1)+8(x-1)(x^2+x+1) is -7. We can see that despite the complex appearance, the expression simplifies to a constant value using the difference of cubes pattern.