\frac{4 x^2-x-34}{x^3-4 x^2-7 x+10}
Answer (1)
Factor the denominator x 3 − 4 x 2 − 7 x + 10 to get ( x − 1 ) ( x + 2 ) ( x − 5 ) .
Factor the numerator 4 x 2 − x − 34 . The roots are irrational, so it cannot be factored nicely with integers.
Write the rational function as ( x − 1 ) ( x + 2 ) ( x − 5 ) 4 x 2 − x − 34 .
Since there are no common factors, the expression is already simplified. The restrictions are x = 1 , − 2 , 5 .
The simplified rational function is ( x − 1 ) ( x + 2 ) ( x − 5 ) 4 x 2 − x − 34 .
Explanation
Problem Analysis We are given the rational function x 3 − 4 x 2 − 7 x + 10 4 x 2 − x − 34 . Our goal is to simplify this expression by factoring both the numerator and the denominator and then canceling any common factors.
Factoring the Denominator Let's start by factoring the denominator, which is a cubic polynomial: x 3 − 4 x 2 − 7 x + 10 . We can look for integer roots by testing factors of the constant term, 10. The factors of 10 are ± 1 , ± 2 , ± 5 , ± 10 . By testing these values, we find that x = 1 , x = − 2 , and x = 5 are roots of the polynomial. Therefore, the denominator can be factored as ( x − 1 ) ( x + 2 ) ( x − 5 ) .
Factoring the Numerator Now, let's factor the numerator, which is a quadratic polynomial: 4 x 2 − x − 34 . We can use the quadratic formula to find the roots of this polynomial. The quadratic formula is given by x = 2 a − b ± b 2 − 4 a c , where a = 4 , b = − 1 , and c = − 34 . Plugging in these values, we get:
x = 2 ( 4 ) 1 ± ( − 1 ) 2 − 4 ( 4 ) ( − 34 ) = 8 1 ± 1 + 544 = 8 1 ± 545 .
Since 545 is not an integer, the roots are irrational, and the quadratic cannot be factored nicely using integers. However, we can express the numerator as 4 ( x − 8 1 + 545 ) ( x − 8 1 − 545 ) .
Simplifying the Rational Function Now we have the factored forms of the numerator and the denominator. The rational function can be written as:
x 3 − 4 x 2 − 7 x + 10 4 x 2 − x − 34 = ( x − 1 ) ( x + 2 ) ( x − 5 ) 4 x 2 − x − 34
Since there are no common factors between the numerator and the denominator, the rational function cannot be simplified further.
Final Answer Therefore, the simplified form of the rational function is ( x − 1 ) ( x + 2 ) ( x − 5 ) 4 x 2 − x − 34 . The restrictions on the domain are x = 1 , x = − 2 , and x = 5 , as these values would make the denominator equal to zero.
Examples
Rational functions are used in various fields, such as physics, engineering, and economics. For example, in physics, they can describe the relationship between distance, rate, and time. In electrical engineering, they can be used to analyze circuits. Understanding how to simplify rational functions helps in modeling and solving real-world problems in these fields. For instance, if you are designing a bridge, you might use rational functions to model the load distribution and ensure the structure's stability. By simplifying these functions, you can more easily analyze the behavior of the bridge under different conditions, ensuring its safety and longevity.
