Divide.

$\left(x^3-6 x^2+2 x+3\right) \div (x-1)$

Perform polynomial long division of x 3 − 6 x 2 + 2 x + 3 by x − 1 .
Divide x 3 by x to get x 2 , multiply ( x − 1 ) by x 2 and subtract from the dividend.
Divide − 5 x 2 by x to get − 5 x , multiply ( x − 1 ) by − 5 x and subtract from the dividend.
Divide − 3 x by x to get − 3 , multiply ( x − 1 ) by − 3 and subtract from the dividend, resulting in a remainder of 0 and the quotient x 2 − 5 x − 3 ​ .

Explanation

Understanding the Problem We are asked to divide the polynomial x 3 − 6 x 2 + 2 x + 3 by the polynomial x − 1 . This can be done using polynomial long division.

First Step of Long Division We set up the long division problem as follows:


x - 1 | x^3 - 6x^2 + 2x + 3

First, we divide the first term of the dividend ( x 3 ) by the first term of the divisor ( x ) to get the first term of the quotient ( x 2 ).
x^2
x - 1 | x^3 - 6x^2 + 2x + 3

Next, we multiply the divisor ( x − 1 ) by the first term of the quotient ( x 2 ) to get x 3 − x 2 .
x^2
x - 1 | x^3 - 6x^2 + 2x + 3
-(x^3 - x^2)

Subtract this result from the dividend to get the new dividend: ( x 3 − 6 x 2 + 2 x + 3 ) − ( x 3 − x 2 ) = − 5 x 2 + 2 x + 3 .
x^2
x - 1 | x^3 - 6x^2 + 2x + 3
-(x^3 - x^2)
-------------
-5x^2 + 2x + 3


Second Step of Long Division Now, we divide the first term of the new dividend ( − 5 x 2 ) by the first term of the divisor ( x ) to get the next term of the quotient ( − 5 x ).

x^2 - 5x
x - 1 | x^3 - 6x^2 + 2x + 3
-(x^3 - x^2)
-------------
-5x^2 + 2x + 3

Multiply the divisor ( x − 1 ) by this term ( − 5 x ) to get − 5 x 2 + 5 x .
x^2 - 5x
x - 1 | x^3 - 6x^2 + 2x + 3
-(x^3 - x^2)
-------------
-5x^2 + 2x + 3
-(-5x^2 + 5x)

Subtract this result from the new dividend to get the next new dividend: ( − 5 x 2 + 2 x + 3 ) − ( − 5 x 2 + 5 x ) = − 3 x + 3 .
x^2 - 5x
x - 1 | x^3 - 6x^2 + 2x + 3
-(x^3 - x^2)
-------------
-5x^2 + 2x + 3
-(-5x^2 + 5x)
-------------
-3x + 3


Third Step of Long Division Next, we divide the first term of the new dividend ( − 3 x ) by the first term of the divisor ( x ) to get the next term of the quotient ( − 3 ).

x^2 - 5x - 3
x - 1 | x^3 - 6x^2 + 2x + 3
-(x^3 - x^2)
-------------
-5x^2 + 2x + 3
-(-5x^2 + 5x)
-------------
-3x + 3

Multiply the divisor ( x − 1 ) by this term ( − 3 ) to get − 3 x + 3 .
x^2 - 5x - 3
x - 1 | x^3 - 6x^2 + 2x + 3
-(x^3 - x^2)
-------------
-5x^2 + 2x + 3
-(-5x^2 + 5x)
-------------
-3x + 3
-(-3x + 3)

Subtract this result from the new dividend to get the remainder: ( − 3 x + 3 ) − ( − 3 x + 3 ) = 0 .
x^2 - 5x - 3
x - 1 | x^3 - 6x^2 + 2x + 3
-(x^3 - x^2)
-------------
-5x^2 + 2x + 3
-(-5x^2 + 5x)
-------------
-3x + 3
-(-3x + 3)
----------
0


Final Result The quotient is x 2 − 5 x − 3 and the remainder is 0 . Therefore, the result of the division is x 2 − 5 x − 3 .

Examples
Polynomial division is a fundamental concept in algebra with numerous real-world applications. For instance, engineers use polynomial division to model and analyze systems, such as control systems, signal processing, and circuit analysis. In computer graphics, polynomial division can be used to perform operations like texture mapping and image scaling. Moreover, in economics, polynomial models are used to analyze cost functions and revenue functions, where division can help determine break-even points and profit margins. Understanding polynomial division provides a versatile tool for solving problems across various disciplines.

Answered by GinnyAnswer | 2025-07-07