Divide the polynomials.
The form of your answer should either be [tex]$p(x)$[/tex] or [tex]$p(x)+\frac{k}{x+1}$[/tex] where [tex]$p(x)$[/tex] is a polynomial and [tex]$k$[/tex] is an integer.
[tex]$\frac{3 x^3+x-11}{x+1}=\square$[/tex]
Answer (2)
Perform polynomial long division of 3 x 3 + x − 11 by x + 1 .
Find the quotient and remainder: 3 x 3 + x − 11 = ( x + 1 ) ( 3 x 2 − 3 x + 4 ) − 15 .
Express the result as a polynomial plus a fraction: x + 1 3 x 3 + x − 11 = 3 x 2 − 3 x + 4 − x + 1 15 .
The final answer is: 3 x 2 − 3 x + 4 − x + 1 15 .
Explanation
Understanding the Problem We are asked to divide the polynomial 3 x 3 + x − 11 by x + 1 and express the result in the form p ( x ) or p ( x ) + x + 1 k , where p ( x ) is a polynomial and k is an integer.
Solution Strategy We will perform polynomial long division to find the quotient and remainder.
Performing Polynomial Division Dividing 3 x 3 + x − 11 by x + 1 , we obtain a quotient of 3 x 2 − 3 x + 4 and a remainder of − 15 . This means that 3 x 3 + x − 11 = ( x + 1 ) ( 3 x 2 − 3 x + 4 ) − 15.
Expressing the Result Therefore, we can write x + 1 3 x 3 + x − 11 = 3 x 2 − 3 x + 4 − x + 1 15 .
Final Result Thus, the result of the division is 3 x 2 − 3 x + 4 − x + 1 15 .
Examples
Polynomial division is a fundamental concept in algebra and has practical applications in various fields. For example, engineers use polynomial division to analyze and design control systems. Imagine designing a cruise control system for a car. The system's behavior can be modeled using polynomials, and polynomial division helps engineers understand how the system responds to different inputs and disturbances, ensuring stable and efficient performance. This ensures the car maintains a consistent speed, providing a smooth and safe driving experience.
To divide the polynomial 3 x 3 + x − 11 by x + 1 , we performed polynomial long division, yielding a quotient of 3 x 2 − 3 x + 4 and a remainder of − 15 . The final answer is expressed as 3 x 2 − 3 x + 4 − x + 1 15 . Thus, the result is 3 x 2 − 3 x + 4 − x + 1 15 .
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