Divide the polynomials.
The form of your answer should either be $p(x)$ or $p(x)+\frac{k}{x+3}$ where $p(x)$ is a polynomial and $k$ is an integer.
$\frac{2 x^3-x^2-25 x-12}{x+3}=$
Answer (2)
Perform polynomial long division of 2 x 3 − x 2 − 25 x − 12 by x + 3 .
The division yields a quotient of 2 x 2 − 7 x − 4 and a remainder of 0 .
Express the result as p ( x ) + x + 3 k , where p ( x ) = 2 x 2 − 7 x − 4 and k = 0 .
The final answer is 2 x 2 − 7 x − 4 . 2 x 2 − 7 x − 4
Explanation
Understanding the Problem We are given the polynomial division problem x + 3 2 x 3 − x 2 − 25 x − 12 . Our goal is to perform this division and express the result in the form p ( x ) or p ( x ) + x + 3 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 − x 2 − 25 x − 12 by x + 3 .
First Step of Division Dividing 2 x 3 by x gives 2 x 2 . Multiplying x + 3 by 2 x 2 gives 2 x 3 + 6 x 2 . Subtracting this from 2 x 3 − x 2 gives − 7 x 2 . Bringing down the next term, we have − 7 x 2 − 25 x .
Second Step of Division Dividing − 7 x 2 by x gives − 7 x . Multiplying x + 3 by − 7 x gives − 7 x 2 − 21 x . Subtracting this from − 7 x 2 − 25 x gives − 4 x . Bringing down the last term, we have − 4 x − 12 .
Third Step of Division Dividing − 4 x by x gives − 4 . Multiplying x + 3 by − 4 gives − 4 x − 12 . Subtracting this from − 4 x − 12 gives 0 .
Final Result The quotient is 2 x 2 − 7 x − 4 and the remainder is 0 . Therefore, the result of the division is 2 x 2 − 7 x − 4 + x + 3 0 = 2 x 2 − 7 x − 4 .
Conclusion Thus, x + 3 2 x 3 − x 2 − 25 x − 12 = 2 x 2 − 7 x − 4 .
Examples
Polynomial division is a fundamental concept in algebra with numerous real-world applications. For instance, engineers use polynomial division when designing systems and analyzing their stability. Imagine designing a control system for a robot arm; the transfer function, which describes the relationship between the input and output, can often be expressed as a ratio of polynomials. Simplifying this ratio using polynomial division helps engineers understand the system's behavior and ensure it operates correctly. This ensures the robot arm moves smoothly and accurately.
To divide the polynomial x + 3 2 x 3 − x 2 − 25 x − 12 , we performed polynomial long division, resulting in a quotient of 2 x 2 − 7 x − 4 with a remainder of 0. Therefore, the answer is simply 2 x 2 − 7 x − 4 .
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