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Simplifying the Expression: (x^2-2x/2x^2+8-2x^2/8-4x+2x^2-x^3)(1-1/x-2/x^2)
This article will guide you through the process of simplifying the complex expression:
(x^2-2x/2x^2+8-2x^2/8-4x+2x^2-x^3)(1-1/x-2/x^2)
Let's break down the steps to achieve a simplified form.
Step 1: Simplifying the First Expression
First, we need to simplify the expression within the first set of parentheses:
(x^2-2x/2x^2+8-2x^2/8-4x+2x^2-x^3)
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Combine like terms:
- (x^2 + 2x^2 - x^3) - (2x/2x^2) - (2x^2/8-4x) + 8
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Simplify fractions:
- (3x^2 - x^3) - (1/x) - (x^2/4-2x) + 8
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Rearrange the terms:
- -x^3 + 3x^2 - x^2/(4-2x) - 1/x + 8
Step 2: Simplifying the Second Expression
Next, we simplify the expression within the second set of parentheses:
(1-1/x-2/x^2)
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Find a common denominator:
- (x^2/x^2 - x/x^2 - 2/x^2)
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Combine terms:
- (x^2-x-2)/x^2
Step 3: Multiplying the Simplified Expressions
Now, we multiply the simplified expressions from Steps 1 and 2:
(-x^3 + 3x^2 - x^2/(4-2x) - 1/x + 8) * (x^2-x-2)/x^2
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Distribute:
- (-x^5 + x^4 + 2x^3 + 3x^4 - 3x^3 - 6x^2 - x^2(x^2-x-2)/x^2(4-2x) + (x+2)/x^3 + 8x^2 - 8x - 16)/x^2
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Simplify by canceling common factors:
- (-x^5 + 4x^4 - x^3 - 6x^2 - (x^2-x-2)/(4-2x) + (x+2)/x^3 + 8x^2 - 8x - 16)/x^2
Step 4: Further Simplification (Optional)
We can simplify the expression further by finding a common denominator for the remaining terms and combining them. However, this process can be quite involved. The expression obtained in Step 3 is already in a relatively simplified form.
Conclusion
By breaking down the original expression into smaller parts and simplifying them systematically, we arrive at a simplified form:
(-x^5 + 4x^4 - x^3 - 6x^2 - (x^2-x-2)/(4-2x) + (x+2)/x^3 + 8x^2 - 8x - 16)/x^2
This expression can be further simplified depending on the desired level of detail and complexity.