Question - Finding the Least Common Denominator (LCD) for Rational Equations
Solution:
To find the least common denominator (LCD) for the given rational equation, we must first determine the denominators of each of the individual fractions in the equation and factor them if necessary.The denominators in the equation are:1. $$x - 2$$2. $$x^2 - x - 2$$3. $$x + 1$$The second denominator $$x^2 - x - 2$$ can be factored to identify its components.Factoring $$x^2 - x - 2$$ we get:$$x^2 - x - 2 = (x - 2)(x + 1)$$Now we have the fully factored denominators:1. $$x - 2$$2. $$(x - 2)(x + 1)$$3. $$x + 1$$The least common denominator (LCD) must contain all factors present in these individual denominators in such a way that each denominator divides the LCD without a remainder. In this case, since $$(x - 2)(x + 1)$$ already contains the other factors $$(x - 2)$$ and $$(x + 1)$$, it serves as the LCD.So the LCD for the given rational equation is:$$(x - 2)(x + 1)$$
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