Divide the polynomials.
Your answer should be in the form [tex]$p(x)+\frac{k}{x+1}$[/tex] where [tex]$p$[/tex] is a polynomial and [tex]$k$[/tex] is an integer.
[tex]$\frac{x^2-2 x-8}{x+1}=$\square[/tex]
Answer (2)
Perform polynomial long division of x 2 − 2 x − 8 by x + 1 .
Divide x 2 by x to get x , then multiply x by ( x + 1 ) and subtract from x 2 − 2 x .
Divide − 3 x by x to get − 3 , then multiply − 3 by ( x + 1 ) and subtract from − 3 x − 8 .
The result is x − 3 + x + 1 − 5 , so the final answer is x − 3 + x + 1 − 5 .
Explanation
Understanding the Problem We are given the rational function x + 1 x 2 − 2 x − 8 and our goal is to express it in the form p ( x ) + x + 1 k , where p ( x ) is a polynomial and k is an integer. This requires us to perform polynomial long division.
Setting up the Division We will perform polynomial long division to divide x 2 − 2 x − 8 by x + 1 .
First Step of Long Division Dividing x 2 by x gives x . Multiply x by ( x + 1 ) to get x 2 + x . Subtract this from x 2 − 2 x to get ( x 2 − 2 x ) − ( x 2 + x ) = − 3 x . Bring down the − 8 to get − 3 x − 8 .
Second Step of Long Division Dividing − 3 x by x gives − 3 . Multiply − 3 by ( x + 1 ) to get − 3 x − 3 . Subtract this from − 3 x − 8 to get ( − 3 x − 8 ) − ( − 3 x − 3 ) = − 5 .
Expressing the Result The quotient is x − 3 and the remainder is − 5 . Therefore, we can write x + 1 x 2 − 2 x − 8 = x − 3 + x + 1 − 5 .
Final Answer Thus, p ( x ) = x − 3 and k = − 5 . The expression is x − 3 + x + 1 − 5 .
Examples
Polynomial division is a fundamental concept in algebra and is used in various real-world applications. For instance, in engineering, when designing a system, you might model the system's behavior using rational functions. Simplifying these rational functions using polynomial division can help in analyzing the system's stability and performance. Similarly, in computer graphics, polynomial division can be used to manipulate curves and surfaces, which are often represented by polynomial equations. By dividing one polynomial by another, you can change the shape or characteristics of the curve or surface, allowing for complex designs and animations.
To divide x + 1 x 2 − 2 x − 8 , perform polynomial long division to find that it equals x − 3 + x + 1 − 5 , where p ( x ) = x − 3 and k = − 5 .
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