Simplify: [tex]$\frac{3}{x^2-3 x-4}-\frac{2}{x+1}$[/tex]
Answer (2)
Factor the denominator of the first fraction: x 2 − 3 x − 4 = ( x − 4 ) ( x + 1 ) .
Rewrite the expression with the factored denominator: ( x − 4 ) ( x + 1 ) 3 − x + 1 2 .
Find the least common denominator (LCD): ( x − 4 ) ( x + 1 ) .
Combine the fractions and simplify: ( x − 4 ) ( x + 1 ) 3 − 2 ( x − 4 ) = ( x − 4 ) ( x + 1 ) 11 − 2 x .
The simplified expression is ( x − 4 ) ( x + 1 ) 11 − 2 x .
Explanation
Understanding the Problem We are asked to simplify the expression x 2 − 3 x − 4 3 − x + 1 2 . To do this, we need to find a common denominator and combine the fractions.
Factoring the Denominator First, let's factor the denominator of the first fraction, x 2 − 3 x − 4 . We are looking for two numbers that multiply to -4 and add to -3. These numbers are -4 and 1. So, we can factor the denominator as ( x − 4 ) ( x + 1 ) .
Rewriting the Expression Now we can rewrite the expression as ( x − 4 ) ( x + 1 ) 3 − x + 1 2 .
Finding the Common Denominator The least common denominator (LCD) of the two fractions is ( x − 4 ) ( x + 1 ) . We need to rewrite the second fraction with this denominator. To do this, we multiply the numerator and denominator of the second fraction by ( x − 4 ) : x + 1 2 = ( x + 1 ) ( x − 4 ) 2 ( x − 4 ) = ( x − 4 ) ( x + 1 ) 2 x − 8
Rewriting with Common Denominator Now we can rewrite the original expression with the common denominator: ( x − 4 ) ( x + 1 ) 3 − ( x − 4 ) ( x + 1 ) 2 x − 8
Combining the Fractions Now we can combine the fractions: ( x − 4 ) ( x + 1 ) 3 − ( 2 x − 8 ) = ( x − 4 ) ( x + 1 ) 3 − 2 x + 8 = ( x − 4 ) ( x + 1 ) 11 − 2 x
Simplifying the Expression The numerator is 11 − 2 x , and the denominator is ( x − 4 ) ( x + 1 ) . We check if the numerator can be factored, but it cannot be factored further. Also, there are no common factors between the numerator and the denominator. Therefore, the simplified expression is: ( x − 4 ) ( x + 1 ) 11 − 2 x
Final Answer Thus, the simplified expression is ( x − 4 ) ( x + 1 ) 11 − 2 x .
Examples
Rational expressions are useful in various fields, such as physics and engineering, where they are used to model complex relationships between variables. For example, in electrical engineering, rational functions can describe the impedance of a circuit as a function of frequency. Simplifying these expressions allows engineers to analyze and design circuits more efficiently. Similarly, in physics, rational expressions can appear in equations describing the motion of objects or the behavior of waves. Simplifying these expressions makes it easier to solve these equations and understand the underlying physical phenomena. Understanding how to manipulate and simplify rational expressions is a fundamental skill that enables students to tackle more advanced problems in these fields.
The simplified expression of x 2 − 3 x − 4 3 − x + 1 2 is ( x − 4 ) ( x + 1 ) 11 − 2 x . This is obtained by factoring, finding a common denominator, and then combining the fractions. The final answer has no common factors and is presented in its simplest form.
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