Find the quotient and remainder when $x^5-x^4+2 x^3+4 x^2-6 x+5$ is divided by $x^3+x-1$.
Select one:
A. quotient: $x^2-x+1$; remainder: $6 x^2-8 x+6$
B. quotient: $x^2+x-1$; remainder: $x^2+x-6$
C. quotient: $x^2-x+6$; remainder: $x^2-8 x+6$
D. quotient: $x^2-x-6$; remainder: $x^2-8 x-6$
E. quotient: $x^2+x-8$; remainder: $6 x^2+x+6$
Answer (2)
Perform polynomial long division.
Divide x 5 − x 4 + 2 x 3 + 4 x 2 − 6 x + 5 by x 3 + x − 1 .
The quotient is x 2 − x + 1 and the remainder is 6 x 2 − 8 x + 6 .
The final answer is quotient: x 2 − x + 1 ; remainder: 6 x 2 − 8 x + 6 . x 2 − x + 1 ; 6 x 2 − 8 x + 6
Explanation
Understanding the Problem We want to find the quotient and remainder when the polynomial x 5 − x 4 + 2 x 3 + 4 x 2 − 6 x + 5 is divided by x 3 + x − 1 . This is a polynomial long division problem.
First Step of Long Division We perform polynomial long division. First, divide x 5 by x 3 to get x 2 . Multiply x 2 by x 3 + x − 1 to get x 5 + x 3 − x 2 . Subtract this from x 5 − x 4 + 2 x 3 + 4 x 2 − 6 x + 5 to get − x 4 + x 3 + 5 x 2 − 6 x + 5 .
Second Step of Long Division Next, divide − x 4 by x 3 to get − x . Multiply − x by x 3 + x − 1 to get − x 4 − x 2 + x . Subtract this from − x 4 + x 3 + 5 x 2 − 6 x + 5 to get x 3 + 6 x 2 − 7 x + 5 .
Third Step of Long Division Next, divide x 3 by x 3 to get 1 . Multiply 1 by x 3 + x − 1 to get x 3 + x − 1 . Subtract this from x 3 + 6 x 2 − 7 x + 5 to get 6 x 2 − 8 x + 6 .
Final Result Since the degree of 6 x 2 − 8 x + 6 is less than the degree of x 3 + x − 1 , we have reached the remainder. Thus, the quotient is x 2 − x + 1 and the remainder is 6 x 2 − 8 x + 6 .
Conclusion Therefore, the quotient is x 2 − x + 1 and the remainder is 6 x 2 − 8 x + 6 .
Examples
Polynomial division is used in various engineering fields, such as control systems and signal processing, to simplify complex transfer functions and analyze system behavior. For example, when designing a filter, engineers use polynomial division to decompose a high-order transfer function into simpler, manageable components, making it easier to implement and analyze the filter's frequency response. This technique allows for efficient design and optimization of systems.
The quotient when dividing x 5 − x 4 + 2 x 3 + 4 x 2 − 6 x + 5 by x 3 + x − 1 is x 2 − x + 1 , and the remainder is 6 x 2 − 8 x + 6 . Therefore, the chosen option is A.
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