Subtract the rational expressions.
$\frac{3 x+1}{x^2+11 x+24}-\frac{2 x-7}{x^2+11 x+24}= \square$ (Simplify your answer.)
Answer (1)
Combine the numerators over the common denominator: x 2 + 11 x + 24 ( 3 x + 1 ) − ( 2 x − 7 ) .
Simplify the numerator: ( 3 x + 1 ) − ( 2 x − 7 ) = x + 8 .
Factor the denominator: x 2 + 11 x + 24 = ( x + 3 ) ( x + 8 ) .
Cancel the common factor ( x + 8 ) : ( x + 3 ) ( x + 8 ) x + 8 = x + 3 1 .
The simplified expression is x + 3 1 .
Explanation
Understanding the Problem We are asked to subtract two rational expressions. Both expressions have the same denominator, which is x 2 + 11 x + 24 . Our goal is to simplify the result after performing the subtraction.
Combining Numerators Since the denominators are the same, we can combine the numerators over the common denominator: x 2 + 11 x + 24 3 x + 1 − x 2 + 11 x + 24 2 x − 7 = x 2 + 11 x + 24 ( 3 x + 1 ) − ( 2 x − 7 ) Now, we simplify the numerator: ( 3 x + 1 ) − ( 2 x − 7 ) = 3 x + 1 − 2 x + 7 = x + 8 So the expression becomes: x 2 + 11 x + 24 x + 8
Factoring the Denominator Next, we factor the denominator x 2 + 11 x + 24 . We are looking for two numbers that multiply to 24 and add up to 11. These numbers are 3 and 8. Therefore, x 2 + 11 x + 24 = ( x + 3 ) ( x + 8 ) Now we rewrite the expression as: ( x + 3 ) ( x + 8 ) x + 8
Canceling Common Factors We can cancel the common factor ( x + 8 ) from the numerator and the denominator, provided that x = − 8 :
( x + 3 ) ( x + 8 ) x + 8 = x + 3 1 So the simplified expression is x + 3 1 .
Final Answer Therefore, the simplified expression is x + 3 1 .
Examples
Rational expressions are useful in various real-world applications, such as calculating the average cost of producing a certain number of items. For example, if the cost of producing x items is given by 3 x + 1 and the revenue is given by 2 x − 7 , then the average profit per item can be expressed as a rational expression. Simplifying such expressions can help in making informed business decisions.
