What is the sum? [tex]$\frac{3}{x^2-9}+\frac{5}{x+3}$[/tex]

Factor the denominators of the given rational expressions.
Find the least common denominator (LCD) for all expressions.
Rewrite each expression with the LCD.
Add the numerators and simplify the resulting rational expression: ( x − 3 ) ( x + 3 ) ( x − 2 ) 5 x 3 − 2 x 2 − 68 x + 96 ​ ​

Explanation

Problem Setup We are asked to find the sum of the following expressions:

x 2 − 9 3 ​ + x + 3 5 ​ + x 2 + x − 6 8 ​ + x − 3 5 x − 12 ​ + ( x + 3 ) ( x − 3 ) − 5 x ​ + ( x + 3 ) ( x − 3 ) 5 x − 12 ​

Factoring Denominators First, let's factor the denominators to identify common denominators and simplify the expressions:

x 2 − 9 = ( x − 3 ) ( x + 3 ) x 2 + x − 6 = ( x + 3 ) ( x − 2 )

Rewriting Expressions Now we rewrite the expressions with the factored denominators:

( x − 3 ) ( x + 3 ) 3 ​ + x + 3 5 ​ + ( x + 3 ) ( x − 2 ) 8 ​ + x − 3 5 x − 12 ​ + ( x + 3 ) ( x − 3 ) − 5 x ​ + ( x + 3 ) ( x − 3 ) 5 x − 12 ​

Finding Common Denominator To add these expressions, we need a common denominator. The least common denominator (LCD) is ( x − 3 ) ( x + 3 ) ( x − 2 ) . We rewrite each fraction with this LCD:

( x − 3 ) ( x + 3 ) ( x − 2 ) 3 ( x − 2 ) ​ + ( x + 3 ) ( x − 3 ) ( x − 2 ) 5 ( x − 3 ) ( x − 2 ) ​ + ( x + 3 ) ( x − 2 ) ( x − 3 ) 8 ( x − 3 ) ​ + ( x − 3 ) ( x + 3 ) ( x − 2 ) ( 5 x − 12 ) ( x + 3 ) ( x − 2 ) ​ + ( x + 3 ) ( x − 3 ) ( x − 2 ) − 5 x ( x − 2 ) ​ + ( x + 3 ) ( x − 3 ) ( x − 2 ) ( 5 x − 12 ) ( x − 2 ) ​

Adding Numerators Now we add the numerators:

( x − 3 ) ( x + 3 ) ( x − 2 ) 3 ( x − 2 ) + 5 ( x − 3 ) ( x − 2 ) + 8 ( x − 3 ) + ( 5 x − 12 ) ( x + 3 ) ( x − 2 ) − 5 x ( x − 2 ) + ( 5 x − 12 ) ( x − 2 ) ​

Expanding Numerator Expanding the numerator:

( x − 3 ) ( x + 3 ) ( x − 2 ) 3 x − 6 + 5 ( x 2 − 5 x + 6 ) + 8 x − 24 + ( 5 x − 12 ) ( x 2 + x − 6 ) − 5 x 2 + 10 x + 5 x 2 − 10 x − 12 x + 24 ​
( x − 3 ) ( x + 3 ) ( x − 2 ) 3 x − 6 + 5 x 2 − 25 x + 30 + 8 x − 24 + 5 x 3 + 5 x 2 − 30 x − 12 x 2 − 12 x + 72 − 5 x 2 + 10 x + 5 x 2 − 22 x + 24 ​

Combining Like Terms Combining like terms in the numerator:

( x − 3 ) ( x + 3 ) ( x − 2 ) 5 x 3 + ( 5 x 2 + 5 x 2 − 12 x 2 − 5 x 2 + 5 x 2 ) + ( 3 x − 25 x + 8 x − 30 x − 12 x + 10 x − 22 x ) + ( − 6 + 30 − 24 + 72 + 24 ) ​
( x − 3 ) ( x + 3 ) ( x − 2 ) 5 x 3 − 2 x 2 − 68 x + 96 ​

Simplified Expression The simplified expression is:

( x − 3 ) ( x + 3 ) ( x − 2 ) 5 x 3 − 2 x 2 − 68 x + 96 ​

Final Answer The final answer is ( x − 3 ) ( x + 3 ) ( x − 2 ) 5 x 3 − 2 x 2 − 68 x + 96 ​

Examples
Understanding how to add rational expressions is crucial in many areas of engineering and physics, especially when dealing with systems that can be modeled using transfer functions. For instance, in electrical engineering, when analyzing circuits with multiple components, the overall behavior can be described by adding individual transfer functions, each representing a component's response. Simplifying these expressions allows engineers to predict the circuit's behavior and optimize its design for specific applications. This skill is also vital in control systems, where rational functions describe the system's response to different inputs, and combining them helps in designing stable and efficient controllers.

Answered by GinnyAnswer | 2025-07-03

The sum of the expressions x 2 − 9 3 ​ + x + 3 5 ​ simplifies to ( x − 3 ) ( x + 3 ) 5 x − 12 ​ . This expression shows how the two fractions combine over a common denominator. Remember that x cannot equal 3 or -3 because these values would make the denominator zero.
;

Answered by Anonymous | 2025-07-04