(x^3+3x^2-6x+2)/(x-1)

5 min read Jun 17, 2024

Polynomial Long Division: (x^3+3x^2-6x+2)/(x-1)

This article explores the process of dividing the polynomial x^3 + 3x^2 - 6x + 2 by x - 1 using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is a method for dividing polynomials, similar to the long division method used for numbers. It helps us to rewrite a polynomial as a product of two polynomials (quotient and divisor) plus a remainder.

Steps for Polynomial Long Division

Let's break down the division of (x^3 + 3x^2 - 6x + 2) by (x - 1):

Set up the division:

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

Divide the leading terms:
- The leading term of the dividend (x^3) is divided by the leading term of the divisor (x).
- The result is x^2, which is placed above the x^2 term in the quotient.
```
     x^2 ______
x - 1 | x^3 + 3x^2 - 6x + 2
```
Multiply the divisor by the quotient term:
- Multiply (x - 1) by x^2, which gives x^3 - x^2.
```
     x^2 ______
x - 1 | x^3 + 3x^2 - 6x + 2
        x^3 - x^2
```
Subtract:
- Subtract the product obtained in the previous step from the dividend.
- Change the signs of the terms in the product and add.
```
     x^2 ______
x - 1 | x^3 + 3x^2 - 6x + 2
        x^3 - x^2
        -------
              4x^2 
```

Bring down the next term:

Bring down the next term of the dividend (-6x).

     x^2 ______
x - 1 | x^3 + 3x^2 - 6x + 2
        x^3 - x^2
        -------
              4x^2 - 6x

Repeat steps 2-5:
- Divide the new leading term (4x^2) by the leading term of the divisor (x), which gives 4x.
- Multiply (x - 1) by 4x to get 4x^2 - 4x.
- Subtract the product, changing signs and adding.
- Bring down the next term (+2).
```
     x^2 + 4x ______
x - 1 | x^3 + 3x^2 - 6x + 2
        x^3 - x^2
        -------
              4x^2 - 6x 
              4x^2 - 4x
              -------
                    -2x + 2
```

Final step:

Divide -2x by x, which gives -2.
Multiply (x - 1) by -2 to get -2x + 2.
Subtract, changing signs and adding.

     x^2 + 4x - 2 _____
x - 1 | x^3 + 3x^2 - 6x + 2
        x^3 - x^2
        -------
              4x^2 - 6x 
              4x^2 - 4x
              -------
                    -2x + 2
                    -2x + 2
                    -------
                         0

Result

Therefore, the division of (x^3 + 3x^2 - 6x + 2) by (x - 1) gives us:

Quotient: x^2 + 4x - 2 Remainder: 0

This can be expressed as:

(x^3 + 3x^2 - 6x + 2) / (x - 1) = x^2 + 4x - 2

Conclusion

Polynomial long division allows us to systematically divide polynomials and express the result as a quotient and a remainder. This method is crucial for simplifying expressions, factoring polynomials, and solving algebraic equations.