Write the rational expression in lowest terms.
[tex]$\frac{x^2-11 x+28}{x^2+4 x-32}$[/tex]
Answer (2)
To write the rational expression x 2 + 4 x − 32 x 2 − 11 x + 28 in lowest terms, we factor both the numerator and denominator. After canceling the common factor ( x − 4 ) , we find that the expression simplifies to x + 8 x − 7 . Thus, the final answer is x + 8 x − 7 .
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Factor the numerator: x 2 − 11 x + 28 = ( x − 4 ) ( x − 7 ) .
Factor the denominator: x 2 + 4 x − 32 = ( x − 4 ) ( x + 8 ) .
Write the rational expression with factored numerator and denominator: ( x − 4 ) ( x + 8 ) ( x − 4 ) ( x − 7 ) .
Cancel the common factor ( x − 4 ) : x + 8 x − 7 .
The rational expression in lowest terms is x + 8 x − 7 .
Explanation
Understanding the Problem We are given the rational expression x 2 + 4 x − 32 x 2 − 11 x + 28 and we want to write it in lowest terms. This means we need to factor both the numerator and the denominator and then cancel any common factors.
Factoring the Numerator First, let's factor the numerator, x 2 − 11 x + 28 . We are looking for two numbers that multiply to 28 and add up to -11. These numbers are -4 and -7. Therefore, we can factor the numerator as ( x − 4 ) ( x − 7 ) .
Factoring the Denominator Next, let's factor the denominator, x 2 + 4 x − 32 . We are looking for two numbers that multiply to -32 and add up to 4. These numbers are 8 and -4. Therefore, we can factor the denominator as ( x − 4 ) ( x + 8 ) .
Writing the Factored Expression Now we can write the rational expression with the factored numerator and denominator: ( x − 4 ) ( x + 8 ) ( x − 4 ) ( x − 7 ) .
Canceling Common Factors We can cancel the common factor of ( x − 4 ) in the numerator and denominator: ( x − 4 ) ( x + 8 ) ( x − 4 ) ( x − 7 ) = x + 8 x − 7 .
Final Answer The rational expression in lowest terms is x + 8 x − 7 .
Examples
Rational expressions are used in various fields, such as physics, engineering, and economics, to model relationships between variables. For example, in physics, they can describe the motion of objects or the behavior of electrical circuits. In economics, they can model supply and demand curves. Simplifying rational expressions helps in understanding these models and making predictions. For instance, if we have a rational expression representing the efficiency of a machine, simplifying it can help us identify the key factors affecting its performance and optimize its design. Suppose the efficiency of a machine is given by x 2 + 4 x − 32 x 2 − 11 x + 28 , where x is a parameter related to the machine's operation. By simplifying this expression to x + 8 x − 7 , we can more easily analyze how the efficiency changes with respect to x .
