What is the quotient of $(x^3-3 x^2+3 x-2) \div(x^2-x+1)$?
Answer (1)
Perform polynomial long division of ( x 3 − 3 x 2 + 3 x − 2 ) by ( x 2 − x + 1 ) .
Divide x 3 by x 2 to get x , then multiply ( x 2 − x + 1 ) by x and subtract from the original polynomial.
Divide − 2 x 2 by x 2 to get − 2 , then multiply ( x 2 − x + 1 ) by − 2 and subtract from the remaining polynomial.
The quotient is x − 2 , and the remainder is 0 . Thus, the answer is x − 2 .
Explanation
Problem Analysis We are given the polynomial division problem ( x 3 − 3 x 2 + 3 x − 2 ) ÷ ( x 2 − x + 1 ) and asked to find the quotient.
Long Division Setup To find the quotient, we can perform polynomial long division. We divide x 3 − 3 x 2 + 3 x − 2 by x 2 − x + 1 .
Performing Long Division Performing the long division, we have:
x - 2
x^2-x+1 | x^3 - 3x^2 + 3x - 2 - (x^3 - x^2 + x) ____________________
-2x^2 + 2x - 2 - (-2x^2 + 2x - 2) ____________________
0
Identifying the Quotient The quotient is x − 2 and the remainder is 0 .
Final Answer Therefore, the quotient of ( x 3 − 3 x 2 + 3 x − 2 ) ÷ ( x 2 − x + 1 ) is x − 2 .
Examples
Polynomial division is used in various engineering and scientific applications, such as control systems, signal processing, and cryptography. For example, in control systems, polynomial division can be used to simplify transfer functions, which describe the relationship between the input and output of a system. By dividing polynomials, engineers can analyze and design controllers to achieve desired system performance.
