Divide $2 y^2+8$ by $2 y+4$. Which expression represents the quotient and remainder?
A. $y-2-\frac{16}{2 y+4}$
B. $y+2-\frac{4}{2 y+4}$
C. $y+2+\frac{4}{2 y+4}$
D. $y-2+\frac{16}{2 y+4}$
Answer (1)
Perform polynomial long division of 2 y 2 + 8 by 2 y + 4 .
The quotient is y − 2 and the remainder is 16 .
Express the result as y − 2 + 2 y + 4 16 .
The final answer is y − 2 + 2 y + 4 16 .
Explanation
Understanding the problem We are asked to divide 2 y 2 + 8 by 2 y + 4 and express the result in the form of quotient and remainder. This can be done using polynomial long division.
Polynomial Long Division Performing polynomial long division of 2 y 2 + 8 by 2 y + 4 :
Performing the Division First, divide 2 y 2 by 2 y to get y . Multiply 2 y + 4 by y to get 2 y 2 + 4 y . Subtract this from 2 y 2 + 0 y + 8 to get − 4 y + 8 .
Next, divide − 4 y by 2 y to get − 2 . Multiply 2 y + 4 by − 2 to get − 4 y − 8 . Subtract this from − 4 y + 8 to get 16 .
So, the quotient is y − 2 and the remainder is 16 .
Expressing the Result Therefore, the expression can be written as: y − 2 + 2 y + 4 16
Examples
Polynomial division is a fundamental concept in algebra and has practical applications in various fields. For example, in engineering, it can be used to analyze the stability of systems. Imagine designing a control system for a robot arm. The transfer function of the system might be a rational function, and using polynomial division, engineers can simplify this function to understand the system's behavior and ensure it operates safely and efficiently. This allows for precise control and avoids unwanted oscillations or instability.
