Final answer:
To simplify the given algebraic expression, factor the denominators to find a common denominator and then combine the fractions. After finding and multiplying by the common denominators, subtract the numerators, and simplify further to obtain the final expression.
Step-by-step explanation:
Simplifying Algebraic Fractions
The given algebraic expression is:
(3)/(x²-x-2) - (2)/(x²+x-6)
To simplify this expression, we need to factor the denominators and find a common denominator so that we can combine the two fractions. First, we factor the quadratic expressions:
- x² - x - 2 can be factored into (x - 2)(x + 1).
- x² + x - 6 can be factored into (x + 3)(x - 2).
The common denominator will be the product of (x - 2), (x + 1), and (x + 3). Now we need to adjust each fraction by multiplying to have this common denominator:
- For (3)/(x²-x-2), multiply by ((x + 3)/(x + 3)) to get (3(x + 3))/((x - 2)(x + 1)(x + 3)).
- For (2)/(x²+x-6), multiply by ((x + 1)/(x + 1)) to get (2(x + 1))/((x - 2)(x + 1)(x + 3)).
Combining the adjusted fractions gives us:
((3(x + 3)) - (2(x + 1)))/((x - 2)(x + 1)(x + 3))
This simplification allows us to subtract the numerators. To further simplify, distribute the numerals through the parenthesis:
- 3(x + 3) becomes 3x + 9.
- 2(x + 1) becomes 2x + 2.
The final simplified expression is:
((3x + 9) - (2x + 2))/((x - 2)(x + 1)(x + 3))/
Which simplifies further to:
(x + 7)/((x - 2)(x + 1)(x + 3))