What is the quotient of $(x^3-3 x^2+5 x-3) \div(x-1)$?
A. $x^2-2 x-3$
B. $x^2+2 x+7$
C. $x^2-3 x+8$
D. $x^2-2 x+3$
Answer (1)
Use synthetic division to divide the polynomial x 3 − 3 x 2 + 5 x − 3 by x − 1 .
Set up the synthetic division table with the coefficients of the polynomial and the root of the divisor.
Perform the synthetic division to find the coefficients of the quotient and the remainder.
The quotient is x 2 − 2 x + 3 , so the final answer is x 2 − 2 x + 3 .
Explanation
Understanding the Problem We are asked to find the quotient of the polynomial division x − 1 x 3 − 3 x 2 + 5 x − 3 . This means we need to divide the polynomial x 3 − 3 x 2 + 5 x − 3 by x − 1 .
Performing Synthetic Division We can use polynomial long division or synthetic division to find the quotient. Let's use synthetic division. We set up the synthetic division as follows:
1 | 1 -3 5 -3
| 1 -2 3
|--------------
1 -2 3 0
We bring down the leading coefficient 1. Then we multiply 1 by 1 and write the result under -3. We add -3 and 1 to get -2. Then we multiply -2 by 1 and write the result under 5. We add 5 and -2 to get 3. Then we multiply 3 by 1 and write the result under -3. We add -3 and 3 to get 0.
Identifying the Quotient The numbers in the bottom row are the coefficients of the quotient and the remainder. The quotient is x 2 − 2 x + 3 and the remainder is 0.
Final Answer Therefore, the quotient of ( x 3 − 3 x 2 + 5 x − 3 ) ÷ ( x − 1 ) is x 2 − 2 x + 3 .
Examples
Polynomial division is used in various engineering and scientific applications, such as control systems, signal processing, and data analysis. For example, in control systems, polynomial division can be used to determine the stability of a system by finding the roots of the characteristic equation. In signal processing, polynomial division is used in filter design to decompose a transfer function into simpler components. Understanding polynomial division helps engineers and scientists analyze and design complex systems.
