$2 x^3+17 x^2+23 x-42 ; x-1$

To divide the polynomial 2 x 3 + 17 x 2 + 23 x − 42 by x − 1 , we can use polynomial long division or synthetic division which both yield the same quotient of 2 x 2 + 19 x + 42 with a remainder of 0. This indicates that x − 1 is a factor of the polynomial.
;

Answered by Anonymous | 2025-07-04

Divide the polynomial 2 x 3 + 17 x 2 + 23 x − 42 by x − 1 using polynomial long division or synthetic division.
Performing the division, we find the quotient to be 2 x 2 + 19 x + 42 and the remainder to be 0.
Therefore, 2 x 3 + 17 x 2 + 23 x − 42 = ( x − 1 ) ( 2 x 2 + 19 x + 42 ) .
The result of the division is 2 x 2 + 19 x + 42 ​ .

Explanation

Understanding the Problem We are asked to divide the polynomial 2 x 3 + 17 x 2 + 23 x − 42 by x − 1 . This can be done using polynomial long division or synthetic division. Since the remainder is 0, it means that x − 1 is a factor of the polynomial.

Performing Polynomial Long Division We can perform polynomial long division as follows:
2x^2 + 19x + 42


x - 1 | 2x^3 + 17x^2 + 23x - 42 - (2x^3 - 2x^2) ------------------ 19x^2 + 23x - (19x^2 - 19x) ------------------ 42x - 42 - (42x - 42) ------------------ 0
So, the quotient is 2 x 2 + 19 x + 42 and the remainder is 0.

Performing Synthetic Division Alternatively, we can use synthetic division. We set up the synthetic division table with the coefficients of the polynomial (2, 17, 23, -42) and the root of the divisor x − 1 , which is 1.





2
17
23
-42



1

2
19
42



---
----
----
-----



2
19
42
0


The last number in the bottom row is the remainder, which is 0. The other numbers are the coefficients of the quotient, which is 2 x 2 + 19 x + 42 .

Final Result Therefore, the result of dividing 2 x 3 + 17 x 2 + 23 x − 42 by x − 1 is 2 x 2 + 19 x + 42 .

Examples
Polynomial division is used in various applications, such as finding the roots of a polynomial, simplifying rational expressions, and solving algebraic equations. For example, in engineering, polynomial division can be used to analyze the stability of a system or to design filters. In computer graphics, it can be used to model curves and surfaces. Understanding polynomial division helps in simplifying complex mathematical models in these fields.

Answered by GinnyAnswer | 2025-07-04