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 to divide x 2 + 5 x + 5 by x + 3 .
Identify the quotient and the 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 , where p ( x ) = x + 2 and k = − 1 .
x + 2 − x + 3 1
Explanation
Understanding the Problem We are given the rational function x + 3 x 2 + 5 x + 5 and asked to express it in the form p ( x ) + x + 3 k , where p ( x ) is a polynomial and k is an integer. This requires us to perform polynomial long division.
Performing Polynomial Long Division To divide x 2 + 5 x + 5 by x + 3 , we perform polynomial long division. We set up the long division as follows:
x + 2
x + 3 | x^2 + 5x + 5
First, we divide x 2 by x to get x . Then we multiply x by ( x + 3 ) to get x 2 + 3 x . Subtracting this from x 2 + 5 x + 5 gives us 2 x + 5 .
x + 2
__________
x + 3 | x^2 + 5x + 5 -(x^2 + 3x) __________
2x + 5
Next, we divide 2 x by x to get 2 . Then we multiply 2 by ( x + 3 ) to get 2 x + 6 . Subtracting this from 2 x + 5 gives us − 1 .
x + 2
__________
x + 3 | x^2 + 5x + 5 -(x^2 + 3x) __________
2x + 5 -(2x + 6) __________
-1
Expressing the Result The quotient is x + 2 and the remainder is − 1 . Therefore, we can write the given rational function as x + 3 x 2 + 5 x + 5 = x + 2 + x + 3 − 1 = x + 2 − x + 3 1 .
Final Answer Thus, we have p ( x ) = x + 2 and k = − 1 . The expression is x + 2 + x + 3 − 1 .
Examples
Polynomial division is used in various engineering and scientific applications, such as control systems design and signal processing. For example, when designing a filter, engineers often need to simplify complex transfer functions, which are ratios of polynomials. By performing polynomial division, they can reduce the order of the transfer function, making it easier to analyze and implement in a real-world system. This simplification helps in understanding the system's behavior and optimizing its performance.
