Solve: [tex](6 x^2+5 x+1) \div(x+2)[/tex]
Answer (2)
Perform polynomial long division of 6 x 2 + 5 x + 1 by x + 2 .
Divide 6 x 2 by x to get 6 x , then subtract ( 6 x ) ( x + 2 ) from 6 x 2 + 5 x + 1 to get − 7 x + 1 .
Divide − 7 x by x to get − 7 , then subtract ( − 7 ) ( x + 2 ) from − 7 x + 1 to get 15 .
The result is 6 x − 7 + x + 2 15 , so the final answer is 6 x − 7 + x + 2 15 .
Explanation
Understanding the Problem We are asked to solve the expression ( 6 x 2 + 5 x + 1 ) ÷ ( x + 2 ) . This means we need to perform polynomial division.
First Step of Long Division We will perform polynomial long division to divide 6 x 2 + 5 x + 1 by x + 2 . First, divide the leading term 6 x 2 by x to get 6 x . Multiply x + 2 by 6 x to get 6 x 2 + 12 x . Subtract this from 6 x 2 + 5 x + 1 to get ( 6 x 2 + 5 x + 1 ) − ( 6 x 2 + 12 x ) = − 7 x + 1 .
Second Step of Long Division Next, divide − 7 x by x to get − 7 . Multiply x + 2 by − 7 to get − 7 x − 14 . Subtract this from − 7 x + 1 to get ( − 7 x + 1 ) − ( − 7 x − 14 ) = 15 .
Final Result Thus, we have 6 x 2 + 5 x + 1 = ( 6 x − 7 ) ( x + 2 ) + 15 . Therefore, x + 2 6 x 2 + 5 x + 1 = 6 x − 7 + x + 2 15 .
Examples
Polynomial division is used in various engineering and scientific applications, such as control systems, signal processing, and circuit analysis. For example, when designing a filter, engineers use polynomial division to simplify transfer functions and analyze the filter's frequency response. This helps them optimize the filter's performance and ensure it meets the desired specifications. Understanding polynomial division allows engineers to model and analyze complex systems effectively.
To divide 6 x 2 + 5 x + 1 by x + 2 , we use polynomial long division to find that the result is 6 x − 7 + x + 2 15 . We first divide the leading terms and subtract repeatedly to simplify the expression. The final answer is 6 x − 7 + x + 2 15 .
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