Divide the polynomials.
The form of your answer should either be [tex]$p(x)$[/tex] or [tex]$p(x)+\frac{k}{x-5}$[/tex] where [tex]$p(x)$[/tex] is a polynomial and [tex]$k$[/tex] is an integer.
[tex]$\frac{2 x^3-11 x^2+25}{x-5}=\square$[/tex]
Answer (2)
Perform polynomial long division of 2 x 3 − 11 x 2 + 25 by x − 5 .
The quotient is 2 x 2 − x − 5 and the remainder is 0 .
Express the result in the form p ( x ) + x − 5 k , where p ( x ) is the quotient and k is the remainder.
Since the remainder is 0, the final answer is 2 x 2 − x − 5 .
Explanation
Understanding the Problem We are asked to divide the polynomial 2 x 3 − 11 x 2 + 25 by x − 5 . The goal is to express the result in the form p ( x ) or p ( x ) + x − 5 k , where p ( x ) is a polynomial and k is an integer.
Setting up Long Division We will perform polynomial long division to divide 2 x 3 − 11 x 2 + 0 x + 25 by x − 5 .
First Step of Division First, divide 2 x 3 by x to get 2 x 2 . Multiply 2 x 2 by ( x − 5 ) to get 2 x 3 − 10 x 2 . Subtract this from 2 x 3 − 11 x 2 to get − x 2 . Bring down the 0 x term to get − x 2 + 0 x .
Second Step of Division Next, divide − x 2 by x to get − x . Multiply − x by ( x − 5 ) to get − x 2 + 5 x . Subtract this from − x 2 + 0 x to get − 5 x . Bring down the 25 term to get − 5 x + 25 .
Third Step of Division Now, divide − 5 x by x to get − 5 . Multiply − 5 by ( x − 5 ) to get − 5 x + 25 . Subtract this from − 5 x + 25 to get 0 .
Final Result The quotient is 2 x 2 − x − 5 and the remainder is 0 . Therefore, the result of the division is 2 x 2 − x − 5 .
Expressing the Answer Since the remainder is 0, the answer is simply the polynomial 2 x 2 − x − 5 .
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 stability of systems or to simplify complex transfer functions. In computer graphics, polynomial division can be used to perform operations on curves and surfaces. Moreover, polynomial division is essential in cryptography for tasks such as error correction and data encryption, ensuring secure communication and data storage.
To divide the polynomial 2 x 3 − 11 x 2 + 25 by x − 5 , we use long division. The result of the division is 2 x 2 − x − 5 with no remainder. Thus, the final answer is 2 x 2 − x − 5 .
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