Divide the polynomials.
The form of your answer should either be $p(x)$ or $p(x)+\frac{k}{x-6}$ where $p(x)$ is a polynomial and $k$ is an integer.
$\frac{2 x^3-13 x^2+9 x-16}{x-6}=$
Answer (2)
Perform polynomial long division of 2 x 3 − 13 x 2 + 9 x − 16 by x − 6 .
Identify the quotient and the remainder from the long division.
Express the result in the form p ( x ) + x − 6 k , where p ( x ) is the quotient and k is the remainder.
The final answer is 2 x 2 − x + 3 + x − 6 2 .
Explanation
Understanding the Problem We are asked to divide the polynomial 2 x 3 − 13 x 2 + 9 x − 16 by x − 6 . The answer should be in the form p ( x ) or p ( x ) + x − 6 k where p ( x ) is a polynomial and k is an integer. Our goal is to perform polynomial division and express the result in the specified format.
Solution Plan We will perform polynomial long division of 2 x 3 − 13 x 2 + 9 x − 16 by x − 6 . The result will be expressed in the form p ( x ) + x − 6 k , where p ( x ) is the quotient and k is the remainder.
Performing Polynomial Division Let's perform the polynomial division. We divide 2 x 3 − 13 x 2 + 9 x − 16 by x − 6 . First, divide 2 x 3 by x to get 2 x 2 . Multiply 2 x 2 by ( x − 6 ) to get 2 x 3 − 12 x 2 . Subtract this from 2 x 3 − 13 x 2 to get − x 2 . Bring down the 9 x to get − x 2 + 9 x . Next, divide − x 2 by x to get − x . Multiply − x by ( x − 6 ) to get − x 2 + 6 x . Subtract this from − x 2 + 9 x to get 3 x . Bring down the − 16 to get 3 x − 16 . Finally, divide 3 x by x to get 3 . Multiply 3 by ( x − 6 ) to get 3 x − 18 . Subtract this from 3 x − 16 to get 2 . Therefore, the quotient is 2 x 2 − x + 3 and the remainder is 2 .
Expressing the Result The result of the polynomial division is 2 x 2 − x + 3 with a remainder of 2 . We can express this as 2 x 2 − x + 3 + x − 6 2 .
Final Answer The final answer is 2 x 2 − x + 3 + x − 6 2 .
Examples
Polynomial division is a fundamental concept in algebra and has practical applications in various fields. For instance, in engineering, it can be used to analyze the behavior of systems modeled by polynomial equations. Consider a scenario where you need to determine the stability of a control system represented by a transfer function, which is a ratio of two polynomials. By performing polynomial division, you can simplify the transfer function and analyze its poles and zeros, which are crucial for assessing the system's stability. This allows engineers to design and optimize control systems for various applications, such as robotics, aerospace, and process control. Polynomial division helps in understanding the underlying dynamics and ensuring the reliable performance of these systems.
The polynomial 2 x 3 − 13 x 2 + 9 x − 16 divided by x − 6 results in a quotient of 2 x 2 − x + 3 with a remainder of 2 . The answer can be expressed as 2 x 2 − x + 3 + x − 6 2 . This shows the result of polynomial long division clearly and concisely.
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