%5E3-x%5E2(x-4)%2B8.jpeg "(x-2)^3-x^2(x-4)+8")
Simplifying and Factoring the Expression: (x-2)³ - x²(x-4) + 8
This article will guide you through the process of simplifying and factoring the expression (x-2)³ - x²(x-4) + 8.
Step 1: Expanding the Expression
Firstly, we need to expand the expression. Let's start by expanding (x-2)³:
(x-2)³ = (x-2)(x-2)(x-2)
We can expand this by multiplying the first two terms and then multiplying the result by the third term:
(x-2)(x-2)(x-2) = (x² - 4x + 4)(x-2)
Next, we expand this product:
(x² - 4x + 4)(x-2) = x³ - 6x² + 12x - 8
Now, let's expand the second term in the original expression, x²(x-4):
x²(x-4) = x³ - 4x²
Finally, we can rewrite the entire expression:
(x-2)³ - x²(x-4) + 8 = (x³ - 6x² + 12x - 8) - (x³ - 4x²) + 8
Step 2: Simplifying the Expression
Now, we can simplify the expression by combining like terms:
(x³ - 6x² + 12x - 8) - (x³ - 4x²) + 8 = -2x² + 12x
Therefore, the simplified expression is -2x² + 12x.
Step 3: Factoring the Expression
We can factor out a -2x from the simplified expression:
-2x² + 12x = -2x(x - 6)
Conclusion
We have successfully simplified and factored the expression (x-2)³ - x²(x-4) + 8. The simplified expression is -2x² + 12x, and the factored expression is -2x(x - 6). This process demonstrates the importance of expanding, simplifying, and factoring expressions to understand their structure and behavior.