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Simplifying Rational Expressions: A Step-by-Step Guide
This article will guide you through simplifying the rational expression:
(x^(2)+3x-4)/(x^(2)+4x+4)*(2x^(2)+4x)/(x^(2)-4x+3)
We'll break down the process into manageable steps, focusing on factoring and canceling common factors.
1. Factoring the Expressions
The first step is to factor each of the four expressions in the given fraction.
- x^(2)+3x-4: This factors into (x+4)(x-1).
- x^(2)+4x+4: This factors into (x+2)(x+2).
- 2x^(2)+4x: We can factor out a 2x, giving us 2x(x+2).
- x^(2)-4x+3: This factors into (x-1)(x-3).
2. Rewriting the Expression with Factored Terms
Now, we can rewrite the original expression with the factored terms:
(x+4)(x-1) / (x+2)(x+2) * 2x(x+2) / (x-1)(x-3)
3. Identifying and Cancelling Common Factors
Notice that we have (x-1) and (x+2) appearing both in the numerator and denominator. These can be canceled out:
(x+4) / (x+2) * 2x / (x-3)
4. Simplifying the Expression
Finally, we can multiply the remaining terms to get the simplified expression:
2x(x+4) / (x+2)(x-3)
This is the simplified form of the original rational expression. It's important to remember that this expression is undefined when x = -2 or x = 3, as these values would result in a denominator of 0.
Therefore, the simplified form of the given rational expression is 2x(x+4) / (x+2)(x-3), where x ≠ -2 and x ≠ 3.