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

5 min read Jun 17, 2024

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

This article will guide you through the process of dividing the polynomial (x^3+3x^2-x+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. The process involves repeatedly dividing the leading term of the dividend by the leading term of the divisor, multiplying the result by the divisor, subtracting the product from the dividend, and bringing down the next term.

Steps to Divide (x^3+3x^2-x+2) by (x-1)

Set up the division:

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

Divide the leading terms: Divide the leading term of the dividend (x^3) by the leading term of the divisor (x), which gives you x^2.
```
      x^2 _______
x-1 | x^3 + 3x^2 - x + 2 
```
Multiply the divisor: Multiply the quotient (x^2) by the divisor (x-1) to get x^3 - x^2.
```
      x^2 _______
x-1 | x^3 + 3x^2 - x + 2 
      x^3 - x^2
```

Subtract: Subtract the product (x^3 - x^2) from the dividend.

      x^2 _______
x-1 | x^3 + 3x^2 - x + 2 
      x^3 - x^2
      -------
          4x^2 - x

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

      x^2 _______
x-1 | x^3 + 3x^2 - x + 2 
      x^3 - x^2
      -------
          4x^2 - x 
          - x

Repeat steps 2-5: Divide the new leading term (4x^2) by the leading term of the divisor (x), which gives you 4x. Multiply 4x by the divisor (x-1) to get 4x^2 - 4x. Subtract this from the previous result. Bring down the next term (+2).
```
      x^2 + 4x _______
x-1 | x^3 + 3x^2 - x + 2 
      x^3 - x^2
      -------
          4x^2 - x 
          - x
          -------
            3x + 2
```

Final step: Divide the leading term of the new dividend (3x) by the leading term of the divisor (x), which gives you 3. Multiply 3 by the divisor (x-1) to get 3x - 3. Subtract this from the previous result.

      x^2 + 4x + 3 _______
x-1 | x^3 + 3x^2 - x + 2 
      x^3 - x^2
      -------
          4x^2 - x 
          - x
          -------
            3x + 2
            3x - 3
            -------
                  5

The Result

The quotient of (x^3+3x^2-x+2) / (x-1) is x^2 + 4x + 3. The remainder is 5.

Conclusion

We can express the result as: (x^3+3x^2-x+2) / (x-1) = x^2 + 4x + 3 + 5/(x-1)

Polynomial long division is a fundamental skill in algebra, used in various applications like factoring polynomials, finding roots, and simplifying complex expressions.