Find the sum of the terms.
[tex]\frac{x}{x^2+3 x+2}+\frac{3}{x+1}[/tex]
The numerator of the simplified sum is
A. [tex]$x + 3$[/tex]
B. [tex]$3x+6$[/tex]
C. [tex]$4x+6$[/tex]
D. [tex]$4x+2$[/tex]
Answer (2)
Factor the denominator of the first term.
Find a common denominator for both terms.
Combine the terms into a single fraction.
Simplify the numerator to find the answer: 4 x + 6 .
Explanation
Problem Analysis We are given the expression x 2 + 3 x + 2 x + x + 1 3 and we want to find the numerator of the simplified sum.
Factoring the Quadratic First, we factor the denominator of the first term: x 2 + 3 x + 2 = ( x + 1 ) ( x + 2 ) .
Rewriting the Expression Now we rewrite the expression as ( x + 1 ) ( x + 2 ) x + x + 1 3 .
Finding a Common Denominator To add the two fractions, we need a common denominator, which is ( x + 1 ) ( x + 2 ) . We rewrite the second term with this common denominator: ( x + 1 ) ( x + 2 ) 3 ( x + 2 ) .
Combining the Terms Now we can combine the two terms: ( x + 1 ) ( x + 2 ) x + 3 ( x + 2 ) .
Simplifying the Numerator Next, we simplify the numerator: x + 3 ( x + 2 ) = x + 3 x + 6 = 4 x + 6 .
Finding the Numerator So the simplified expression is ( x + 1 ) ( x + 2 ) 4 x + 6 . The numerator of the simplified sum is 4 x + 6 .
Examples
Understanding how to simplify rational expressions is useful in many areas of math and science. For example, in physics, you might use rational expressions to describe the relationship between distance, rate, and time. Simplifying these expressions can make it easier to solve for unknown variables. Also, in engineering, rational expressions can be used to model various systems, and simplifying them can help engineers to analyze and optimize these systems.
The simplified sum of the expression is ( x + 1 ) ( x + 2 ) 4 x + 6 , and the numerator is 4 x + 6 . Therefore, the chosen option is C. 4 x + 6 .
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