Use long division to divide. Specify the quotient and the remainder.

[tex](x^2+9 x-1) \div(x-1)[/tex]

quotient $\square$
remainder $\square$

Using polynomial long division, the quotient when dividing x 2 + 9 x − 1 by x − 1 is x + 10 and the remainder is 9 .
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Answered by Anonymous | 2025-07-04

Perform polynomial long division of x 2 + 9 x − 1 by x − 1 .
Divide x 2 by x to get x , multiply x − 1 by x to get x 2 − x , and subtract from x 2 + 9 x − 1 to get 10 x − 1 .
Divide 10 x by x to get 10 , multiply x − 1 by 10 to get 10 x − 10 , and subtract from 10 x − 1 to get 9 .
The quotient is x + 10 and the remainder is 9 , so the answer is x + 10 ​ and 9 ​ .

Explanation

Problem Analysis We are asked to perform polynomial long division to divide x 2 + 9 x − 1 by x − 1 and determine the quotient and remainder.

Performing Long Division We set up the long division as follows:
x + 10

x - 1 | x^2 + 9x - 1 -(x^2 - x) ---------- 10x - 1 -(10x - 10) ---------- 9


So, when we divide x 2 + 9 x − 1 by x − 1 , we get a quotient of x + 10 and a remainder of 9 .

Finding Quotient and Remainder The quotient is x + 10 and the remainder is 9 .

Examples
Polynomial long division is a fundamental technique used in various fields, such as engineering and computer science. For example, when designing control systems, engineers often need to simplify complex transfer functions, which are ratios of polynomials. Long division helps break down these functions into simpler terms, making the analysis and design of control systems more manageable. Similarly, in computer science, polynomial division is used in coding theory for error detection and correction.

Answered by GinnyAnswer | 2025-07-04