Divide the polynomials.
Your answer should be in the form [tex]$p(x)+\frac{k}{x+3}$[/tex] where [tex]$p$[/tex] is a polynomial and [tex]$k$[/tex] is an integer.
[tex]$\frac{x^2+5 x+5}{x+3}=$\square[/tex]
Answer (1)
Perform polynomial long division of x 2 + 5 x + 5 by x + 3 .
Identify the quotient and remainder from the long division.
Express the result in the form p ( x ) + x + 3 k .
The final answer is x + 2 − x + 3 1 , so x + 2 − x + 3 1 .
Explanation
Understanding the Problem We are given the rational function x + 3 x 2 + 5 x + 5 and we want to express it in the form p ( x ) + x + 3 k , where p ( x ) is a polynomial and k is an integer. This means we need to perform polynomial long division.
Performing Polynomial Long Division We will perform polynomial long division to divide x 2 + 5 x + 5 by x + 3 .
x + 2
x+3 | x^2 + 5x + 5 -(x^2 + 3x) ---------- 2x + 5 -(2x + 6) ---------- -1
The quotient is x + 2 and the remainder is − 1 .
Expressing the Result Therefore, we can write x + 3 x 2 + 5 x + 5 = x + 2 + x + 3 − 1 = x + 2 − x + 3 1 .
Final Answer Thus, p ( x ) = x + 2 and k = − 1 . The expression is x + 2 + x + 3 − 1 .
Examples
Polynomial division is a fundamental concept in algebra and is used in various applications, such as simplifying complex rational expressions, solving equations, and analyzing the behavior of functions. For example, in engineering, polynomial division can be used to model the trajectory of a projectile or to analyze the stability of a control system. By expressing a rational function in the form p ( x ) + x + 3 k , we can better understand its properties and behavior, such as its asymptotes and limits. This technique is also useful in calculus for integrating rational functions.
