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]$\frac{x^2+4}{x-3}=\square$[/tex]
Answer (1)
Perform polynomial long division of x 2 + 4 by x − 3 .
Identify the quotient p ( x ) and the remainder k from the division.
Express the result in the form p ( x ) + x − 3 k .
The final answer is x + 3 + x − 3 13 , so x + 3 + x − 3 13 .
Explanation
Understanding the Problem We are asked to divide the polynomial x 2 + 4 by x − 3 and express the result 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 division.
Performing Polynomial Division We perform polynomial long division to divide x 2 + 4 by x − 3 . The quotient will be our p ( x ) and the remainder will be k .
Result of Division When we divide x 2 + 4 by x − 3 , we obtain a quotient of x + 3 and a remainder of 13 . This can be verified as follows:
( x − 3 ) ( x + 3 ) + 13 = x 2 − 9 + 13 = x 2 + 4
Thus, we have p ( x ) = x + 3 and k = 13 .
Final Answer Therefore, the expression x − 3 x 2 + 4 can be written as x + 3 + x − 3 13 .
Examples
Polynomial division is a fundamental concept in algebra and is used in various applications. For example, engineers use polynomial division to analyze the stability of control systems. Suppose an engineer is designing a control system with a transfer function s − 3 s 2 + 4 , where s is a complex variable. To understand the system's behavior, the engineer might perform polynomial division to rewrite the transfer function as s + 3 + s − 3 13 . This form helps in identifying poles and zeros, which are crucial for determining the system's stability and response characteristics. This decomposition allows for easier analysis and design of the control system.
