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

Jasonbradley

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Jun 17, 2024

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Long Division of Polynomials: (x³ + 3x² - x + 2) / (x - 1)

Long division of polynomials is a method used to divide one polynomial by another. It is similar to the long division of numbers, but with some key differences.

Steps Involved

Let's divide the polynomial (x³ + 3x² - x + 2) by (x - 1) using long division:

1. Set up the division: Write the dividend (x³ + 3x² - x + 2) inside the division symbol and the divisor (x - 1) outside.

             ________
    x - 1 | x³ + 3x² - x + 2

2. Divide the leading terms: Divide the leading term of the dividend (x³) by the leading term of the divisor (x). This gives us x². Write x² above the division symbol.

             x²_______
    x - 1 | x³ + 3x² - x + 2

3. Multiply the divisor by the quotient term: Multiply (x - 1) by x². This gives us x³ - x². Write this result below the dividend.

             x²_______
    x - 1 | x³ + 3x² - x + 2
             x³ - x² 

4. Subtract: Subtract the result from the dividend. Change the signs of the terms in the second row and add.

             x²_______
    x - 1 | x³ + 3x² - x + 2
             x³ - x² 
             -------
                  4x² - x

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

             x²_______
    x - 1 | x³ + 3x² - x + 2
             x³ - x² 
             -------
                  4x² - x + 2

6. Repeat steps 2-5: Repeat the process with the new dividend (4x² - x + 2). Divide the leading term (4x²) by the leading term of the divisor (x), which gives us 4x. Write 4x next to the x² above the division symbol.

             x² + 4x _____
    x - 1 | x³ + 3x² - x + 2
             x³ - x² 
             -------
                  4x² - x + 2
                  4x² - 4x

7. Subtract again: Subtract the result from the new dividend.

             x² + 4x _____
    x - 1 | x³ + 3x² - x + 2
             x³ - x² 
             -------
                  4x² - x + 2
                  4x² - 4x
                  -------
                        3x + 2

8. Bring down the next term: Bring down the last term of the dividend (2).

             x² + 4x _____
    x - 1 | x³ + 3x² - x + 2
             x³ - x² 
             -------
                  4x² - x + 2
                  4x² - 4x
                  -------
                        3x + 2

9. Repeat steps 2-5: Divide the leading term of the new dividend (3x) by the leading term of the divisor (x), which gives us 3. Write 3 next to the 4x above the division symbol.

             x² + 4x + 3 ___
    x - 1 | x³ + 3x² - x + 2
             x³ - x² 
             -------
                  4x² - x + 2
                  4x² - 4x
                  -------
                        3x + 2
                        3x - 3

10. Subtract again: Subtract the result from the new dividend.

             x² + 4x + 3 ___
    x - 1 | x³ + 3x² - x + 2
             x³ - x² 
             -------
                  4x² - x + 2
                  4x² - 4x
                  -------
                        3x + 2
                        3x - 3
                        -----
                              5

11. The remainder: The final result is 5, which is our remainder.

Result

Therefore, the result of dividing (x³ + 3x² - x + 2) by (x - 1) is:

x² + 4x + 3 + 5/(x - 1)

This can also be written as:

x² + 4x + 3 + (5/x-1)