Divide the following polynomials. Then place the answer. Do not include parentheses in your answer. [tex]$\left(2 a^2+a+3\right) \div(a-1)$[/tex]
Answer (2)
Divide 2 a 2 by a to get 2 a , and multiply ( a − 1 ) by 2 a to get 2 a 2 − 2 a .
Subtract 2 a 2 − 2 a from 2 a 2 + a + 3 to get 3 a + 3 .
Divide 3 a by a to get 3 , and multiply ( a − 1 ) by 3 to get 3 a − 3 .
Subtract 3 a − 3 from 3 a + 3 to get a remainder of 6 . The final answer is 2 a + 3 + a − 1 6 .
Explanation
Understanding the Problem We are asked to divide the polynomial 2 a 2 + a + 3 by the polynomial a − 1 . The problem asks us to perform polynomial division and express the result without parentheses.
Solution Plan We will perform polynomial long division to divide 2 a 2 + a + 3 by a − 1 .
First Step of Long Division First, we divide 2 a 2 by a , which gives 2 a . Then, we multiply ( a − 1 ) by 2 a to get 2 a 2 − 2 a . Subtracting this from 2 a 2 + a + 3 , we have ( 2 a 2 + a + 3 ) − ( 2 a 2 − 2 a ) = 3 a + 3 .
Second Step of Long Division Next, we divide 3 a by a , which gives 3 . Then, we multiply ( a − 1 ) by 3 to get 3 a − 3 . Subtracting this from 3 a + 3 , we have ( 3 a + 3 ) − ( 3 a − 3 ) = 6 .
Final Result Therefore, the quotient is 2 a + 3 and the remainder is 6 . So, the result of the division is 2 a + 3 + a − 1 6 .
Conclusion The result of dividing 2 a 2 + a + 3 by a − 1 is 2 a + 3 + a − 1 6 .
Examples
Polynomial division is a fundamental concept in algebra and has practical applications in various fields. For example, in engineering, it can be used to analyze the behavior of systems described by polynomial equations. In computer graphics, polynomial division can be used to manipulate curves and surfaces. Moreover, polynomial division helps in simplifying complex expressions, making them easier to analyze and solve. Understanding polynomial division enhances problem-solving skills and provides a foundation for more advanced mathematical concepts.
The division of the polynomial 2 a 2 + a + 3 by a − 1 results in a quotient of 2 a + 3 and a remainder of 6 , expressed as 2 a + 3 + a − 1 6 .
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