%2F(x%5E2%2B4x%2B4)-6x%2F(x%5E2-4)%2B3%2F(x-2).jpeg "(2x+1)/(x^2+4x+4)-6x/(x^2-4)+3/(x-2)")
Simplifying the Expression: (2x+1)/(x^2+4x+4)-6x/(x^2-4)+3/(x-2)
This article will guide you through the process of simplifying the given algebraic expression:
(2x+1)/(x^2+4x+4)-6x/(x^2-4)+3/(x-2)
Step 1: Factor the denominators
- x² + 4x + 4 can be factored as (x+2)²
- x² - 4 can be factored as (x+2)(x-2)
Now, the expression becomes:
(2x+1)/(x+2)² - 6x/(x+2)(x-2) + 3/(x-2)
Step 2: Find the Least Common Multiple (LCM)
The LCM of the denominators is (x+2)²(x-2)
Step 3: Adjust each fraction to have the LCM as its denominator
- For the first term, we need to multiply the numerator and denominator by (x-2).
- For the second term, we need to multiply the numerator and denominator by (x+2).
- The third term needs to be multiplied by (x+2)².
This gives us:
[(2x+1)(x-2)]/(x+2)²(x-2) - [6x(x+2)]/(x+2)²(x-2) + [3(x+2)²]/(x+2)²(x-2)
Step 4: Combine the numerators
(2x² - 3x - 2 - 6x² - 12x + 3x² + 12x + 12)/(x+2)²(x-2)
Step 5: Simplify the numerator
(-x² - 3x + 10)/(x+2)²(x-2)
Step 6: Factor the numerator (if possible)
The numerator can be factored as (-1)(x-2)(x+5)
Step 7: Simplify the expression
The final simplified expression is:
(-1)(x-2)(x+5)/(x+2)²(x-2)
Final Answer:
-(x+5)/(x+2)²
Important Note: This expression is undefined when x = -2 and x = 2, as these values would make the denominator equal to zero.