What is the difference?
$\frac{2 x+5}{x^2-3 x}-\frac{3 x+5}{x^3-9 x}-\frac{x+1}{x^2-9}$
$\frac{(x+5)(x+2)}{x^3-9 x}$
$\frac{(x+5)(x+4)}{x^3-9 x}$
$\frac{-2 x+11}{x^3-12 x-9}$
$\frac{3(x+2)}{x^2-3 x}$
Answer (1)
Factor the denominators of the first three expressions.
Find a common denominator and combine the first three expressions: x 2 − 3 x 2 x + 5 − x 3 − 9 x 3 x + 5 − x 2 − 9 x + 1 = x 3 − 9 x ( x + 2 ) ( x + 5 ) .
Compare the simplified expression with the given expressions.
The simplified expression is equal to x 3 − 9 x ( x + 5 ) ( x + 2 ) , which is expression 4. Therefore, the difference is x 3 − 9 x ( x + 5 ) ( x + 2 ) .
Explanation
Problem Analysis First, let's analyze the given expressions and identify the math problem we're dealing with. We have a combination of rational expressions involving polynomial terms. Our goal is to simplify the first three rational expressions and then compare the result with the remaining expressions to find the difference.
Factoring Denominators Let's start by factoring the denominators of the first three expressions:
Expression 1: x 2 − 3 x = x ( x − 3 ) Expression 2: x 3 − 9 x = x ( x 2 − 9 ) = x ( x − 3 ) ( x + 3 ) Expression 3: x 2 − 9 = ( x − 3 ) ( x + 3 )
Finding Common Denominator Now, let's find a common denominator for the first three expressions. The common denominator is x ( x − 3 ) ( x + 3 ) . We will rewrite each expression with this common denominator:
Expression 1: x ( x − 3 ) 2 x + 5 = x ( x − 3 ) ( x + 3 ) ( 2 x + 5 ) ( x + 3 ) = x ( x − 3 ) ( x + 3 ) 2 x 2 + 6 x + 5 x + 15 = x ( x − 3 ) ( x + 3 ) 2 x 2 + 11 x + 15 Expression 2: x ( x − 3 ) ( x + 3 ) 3 x + 5 (already has the common denominator) Expression 3: ( x − 3 ) ( x + 3 ) x + 1 = x ( x − 3 ) ( x + 3 ) ( x + 1 ) x = x ( x − 3 ) ( x + 3 ) x 2 + x
Combining and Simplifying Now, let's combine the first three expressions:
x ( x − 3 ) ( x + 3 ) 2 x 2 + 11 x + 15 − x ( x − 3 ) ( x + 3 ) 3 x + 5 − x ( x − 3 ) ( x + 3 ) x 2 + x = x ( x − 3 ) ( x + 3 ) ( 2 x 2 + 11 x + 15 ) − ( 3 x + 5 ) − ( x 2 + x )
Simplify the numerator:
2 x 2 + 11 x + 15 − 3 x − 5 − x 2 − x = ( 2 x 2 − x 2 ) + ( 11 x − 3 x − x ) + ( 15 − 5 ) = x 2 + 7 x + 10
So, the combined expression is:
x ( x − 3 ) ( x + 3 ) x 2 + 7 x + 10
Now, factor the numerator:
x 2 + 7 x + 10 = ( x + 2 ) ( x + 5 )
Thus, the simplified expression is:
x ( x − 3 ) ( x + 3 ) ( x + 2 ) ( x + 5 ) = x 3 − 9 x ( x + 2 ) ( x + 5 )
Comparison Now, let's compare this simplified expression with the given expressions:
Expression 4: x 3 − 9 x ( x + 5 ) ( x + 2 ) Expression 5: x 3 − 9 x ( x + 5 ) ( x + 4 ) Expression 6: x 3 − 12 x − 9 − 2 x + 11 Expression 7: x 2 − 3 x 3 ( x + 2 ) = x ( x − 3 ) 3 ( x + 2 )
We can see that the simplified expression is equal to Expression 4.
Calculating Differences Therefore, the difference between the first three expressions and Expression 4 is 0.
Now let's compare the simplified expression with Expression 5: x 3 − 9 x ( x + 2 ) ( x + 5 ) − x 3 − 9 x ( x + 5 ) ( x + 4 ) = x 3 − 9 x ( x + 5 ) ( x + 2 − ( x + 4 )) = x 3 − 9 x ( x + 5 ) ( − 2 ) = x 3 − 9 x − 2 ( x + 5 )
Now let's compare the simplified expression with Expression 6: x 3 − 9 x ( x + 2 ) ( x + 5 ) − x 3 − 12 x − 9 − 2 x + 11 . These expressions are not easily comparable.
Now let's compare the simplified expression with Expression 7: x ( x − 3 ) ( x + 3 ) ( x + 2 ) ( x + 5 ) − x ( x − 3 ) 3 ( x + 2 ) = x ( x − 3 ) ( x + 3 ) ( x + 2 ) ( x + 5 ) − x ( x − 3 ) ( x + 3 ) 3 ( x + 2 ) ( x + 3 ) = x ( x − 3 ) ( x + 3 ) ( x + 2 ) ( x + 5 − 3 ( x + 3 )) = x ( x − 3 ) ( x + 3 ) ( x + 2 ) ( x + 5 − 3 x − 9 ) = x ( x − 3 ) ( x + 3 ) ( x + 2 ) ( − 2 x − 4 ) = x ( x − 3 ) ( x + 3 ) − 2 ( x + 2 ) ( x + 2 ) = x ( x − 3 ) ( x + 3 ) − 2 ( x + 2 ) 2
Final Answer The question asks for 'the difference', which is ambiguous. However, based on the given options, it seems the question is asking which expression is equivalent to the simplified form of the first three expressions. Since the simplified form of the first three expressions is equal to expression 4, the answer is expression 4.
Simplified Expression The simplified expression is x 3 − 9 x ( x + 5 ) ( x + 2 ) .
Examples
Rational expressions are useful in various real-world applications, such as calculating the average cost of production in economics or determining the concentration of a substance in chemistry. For instance, if a company has a fixed cost and a variable cost that depends on the number of units produced, the average cost can be expressed as a rational expression. Simplifying such expressions helps in analyzing cost trends and making informed business decisions. Suppose a company has a fixed cost of $1000 and a variable cost of 2 x per unit, where x is the number of units produced. The average cost per unit is given by x 1000 + 2 x . Simplifying and analyzing this expression can help the company understand how the average cost changes as production increases.
