-(x-2%2B10-x%5E2%2Fx%2B2).jpeg "(x^2/x^3-4x+6/6-3x+1/x+2) (x-2+10-x^2/x+2)")
Simplifying Complex Algebraic Expressions
This article will guide you through the process of simplifying the complex algebraic expression:
(x^2/x^3-4x+6/6-3x+1/x+2) (x-2+10-x^2/x+2)
Step 1: Simplifying Individual Fractions
First, we need to simplify each individual fraction within the expression:
- x^2/x^3: This simplifies to 1/x.
- 6/6: This simplifies to 1.
- 1/x+2: This remains as 1/(x+2).
- 10-x^2/x+2: This remains as (10-x^2)/(x+2).
Now, the expression becomes:
(1/x-4x+1-3x+1/(x+2)) (x-2+(10-x^2)/(x+2))
Step 2: Combining Like Terms
Next, we combine like terms within the parentheses:
- (1/x - 4x - 3x + 1) (x - 2 + (10 - x^2)/(x + 2))
- (1/x - 7x + 1) (x - 2 + (10 - x^2)/(x + 2))
Step 3: Finding a Common Denominator
To further simplify, we need to find a common denominator for the fractions within each set of parentheses:
- (1/x - 7x^2/x + x/x) (x - 2 + (10 - x^2)/(x + 2))
- ((1 - 7x^2 + x)/x) ((x(x + 2))/(x + 2) - 2(x + 2)/(x + 2) + (10 - x^2)/(x + 2))
Step 4: Combining Numerators
Now, we can combine the numerators over the common denominators:
- ((1 - 7x^2 + x)/x) ((x^2 + 2x - 2x - 4 + 10 - x^2)/(x + 2))
- ((1 - 7x^2 + x)/x) (6/(x + 2))
Step 5: Multiplying the Fractions
Finally, we multiply the two fractions together:
- (1 - 7x^2 + x)(6) / (x(x + 2))
- (6 - 42x^2 + 6x) / (x^2 + 2x)
Final Simplified Expression
The simplified form of the original expression is:
(6 - 42x^2 + 6x) / (x^2 + 2x)
Important Note: This expression can be further simplified by factoring out a 6 from the numerator, but it is not necessary for this solution.