-(1%2Fx%2Bx-2).jpeg "(1/x^2+x-2-x/x+1) (1/x+x-2)")
Simplifying the Expression: (1/x^2+x-2-x/x+1) (1/x+x-2)
This article will walk through the process of simplifying the algebraic expression: (1/x^2+x-2-x/x+1) (1/x+x-2).
Step 1: Factoring the Denominators
The first step is to factor the denominators of the fractions within the expression.
- x^2 + x - 2 factors into (x+2)(x-1)
- x + 1 remains as (x+1)
- x + x - 2 simplifies to 2x - 2 which factors into 2(x-1)
Now, the expression looks like this:
(1/(x+2)(x-1) - x/(x+1)) * (1/2(x-1))
Step 2: Finding a Common Denominator
To combine the fractions within the first set of parentheses, we need a common denominator. The least common denominator is (x+2)(x-1)(x+1).
- Multiply the first fraction by (x+1)/(x+1): (x+1)/(x+2)(x-1)(x+1)
- Multiply the second fraction by (x+2)(x-1)/(x+2)(x-1): -x(x+2)(x-1)/(x+2)(x-1)(x+1)
The expression now becomes:
((x+1) - x(x+2)(x-1))/(x+2)(x-1)(x+1) * (1/2(x-1))
Step 3: Simplifying the Numerator
Let's simplify the numerator:
(x+1 - x(x^2 + x - 2))/(x+2)(x-1)(x+1) * (1/2(x-1))
(x+1 - x^3 - x^2 + 2x)/(x+2)(x-1)(x+1) * (1/2(x-1))
(-x^3 - x^2 + 3x + 1)/(x+2)(x-1)(x+1) * (1/2(x-1))
Step 4: Multiplying the Fractions
Now we can multiply the two fractions together:
(-x^3 - x^2 + 3x + 1)/(x+2)(x-1)(x+1) * 1/2(x-1)
(-x^3 - x^2 + 3x + 1)/(2(x+2)(x-1)^2(x+1))
Conclusion
The simplified form of the expression (1/x^2+x-2-x/x+1) (1/x+x-2) is (-x^3 - x^2 + 3x + 1)/(2(x+2)(x-1)^2(x+1)). It's important to note that this expression is undefined when x = -2, x = 1, or x = -1 due to the presence of these values in the denominator.