Which one of the following is true when [tex]f(x)=x^3+5 x^2-x+8[/tex] divided by [tex]g(x)=x^2+2[/tex]?
A. The degree of the remainder is 1.
B. The remainder is [tex]x+5[/tex]
C. The quotient is [tex]3 x+2[/tex].
D. The constant term of the quotient is -2.
Answer (2)
Perform polynomial long division of f ( x ) = x 3 + 5 x 2 − x + 8 by g ( x ) = x 2 + 2 .
Identify the quotient as x + 5 and the remainder as − 3 x − 2 .
Check the degree of the remainder, which is 1.
Conclude that the correct option is A: The degree of the remainder is 1. A
Explanation
Problem Analysis We are given the polynomial f ( x ) = x 3 + 5 x 2 − x + 8 and we want to divide it by g ( x ) = x 2 + 2 . Our goal is to find the quotient and remainder of this division and then check which of the given options is true.
Polynomial Long Division We will perform polynomial long division to divide f ( x ) by g ( x ) .
Performing the Division
\t\t\t\t x 2 + 2 | x 3 + 5 x 2 − x + 8 \t\t\t\t\t\t\t x + 5 \t\t\t\t\t\t ----------------- \t\t\t\t\t\t x 3 + 0 x 2 + 2 x \t\t\t\t\t\t ----------------- \t\t\t\t\t\t\t 5 x 2 − 3 x + 8 \t\t\t\t\t\t\t 5 x 2 + 0 x + 10 \t\t\t\t\t\t ----------------- \t\t\t\t\t\t\t − 3 x − 2
So, when we divide x 3 + 5 x 2 − x + 8 by x 2 + 2 , the quotient is x + 5 and the remainder is − 3 x − 2 .
Analyzing the Options Now, let's analyze the given options:
A. The degree of the remainder is 1. The remainder is − 3 x − 2 , which has a degree of 1. So, this statement is true.
B. The remainder is x + 5 . The remainder is − 3 x − 2 , so this statement is false.
C. The quotient is 3 x + 2 . The quotient is x + 5 , so this statement is false.
D. The constant term of the quotient is -2. The quotient is x + 5 , so the constant term is 5, and this statement is false.
Final Answer Therefore, the correct answer is A. The degree of the remainder is 1.
Examples
Polynomial division is used in various engineering and scientific applications, such as control systems design, signal processing, and cryptography. For example, in control systems, polynomial division can be used to simplify transfer functions, which describe the relationship between the input and output of a system. By dividing the numerator and denominator polynomials of the transfer function, engineers can obtain a simpler representation of the system's behavior, making it easier to analyze and design controllers.
The correct answer is A: The degree of the remainder is 1. After performing polynomial long division of f ( x ) by g ( x ) , the quotient is x + 5 and the remainder is − 3 x − 2 . Hence, the degree of the remainder is indeed 1.
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