(x^3+5x^2-3x-1)/(x-1) Long Division

Jasonbradley

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Jun 17, 2024

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Long Division of Polynomials: (x^3 + 5x^2 - 3x - 1) / (x - 1)

Long division of polynomials is a technique used to divide a polynomial by another polynomial of lower or equal degree. It is similar to the long division of numbers you learned in elementary school.

In this example, we will divide the polynomial x^3 + 5x^2 - 3x - 1 by the polynomial x - 1.

Steps to Perform Long Division:

1. Set up the division: Write the dividend (x^3 + 5x^2 - 3x - 1) inside the division symbol and the divisor (x - 1) outside.

        ________
   x - 1 | x^3 + 5x^2 - 3x - 1
   

2. Divide the leading terms: Divide the leading term of the dividend (x^3) by the leading term of the divisor (x). This gives you x^2, which you write above the division symbol.

        x^2 ______
   x - 1 | x^3 + 5x^2 - 3x - 1
   

3. Multiply and subtract: Multiply the divisor (x - 1) by the term you just wrote (x^2). This gives you x^3 - x^2. Subtract this product from the dividend.

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

4. Bring down the next term: Bring down the next term of the dividend (-3x).

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

5. Repeat steps 2-4: Now divide the leading term of the new dividend (6x^2) by the leading term of the divisor (x). This gives you 6x, which you write above the division symbol.

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

6. Continue until the degree of the remainder is less than the degree of the divisor: Repeat steps 2-4 until the degree of the remainder is less than the degree of the divisor.

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

Result:

The quotient is x^2 + 6x + 3 and the remainder is 2.

Therefore, (x^3 + 5x^2 - 3x - 1) / (x - 1) = x^2 + 6x + 3 + 2/(x - 1)