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

Jasonbradley

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Jun 17, 2024

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

Long division of polynomials is a method used to divide one polynomial by another polynomial. Here's how to divide (x^3 + 3x^2 - x - 3) by (x - 1):

Steps

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

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

2. Divide the leading terms: Divide the leading term of the dividend (x^3) by the leading term of the divisor (x). This gives us x^2. Write this above the division symbol.

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

3. Multiply the quotient by the divisor: Multiply the quotient (x^2) by the divisor (x - 1). This gives us x^3 - x^2. Write this below the dividend, aligning terms with corresponding powers of x.

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

4. Subtract: Subtract the product (x^3 - x^2) from the dividend. Remember to change the signs of the terms in the product before subtracting.

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

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

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

6. Repeat steps 2-5: Divide the leading term of the new dividend (4x^2) by the leading term of the divisor (x). This gives us 4x. Write this above the division symbol.

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

   Multiply the new quotient (4x) by the divisor (x - 1), which gives us 4x^2 - 4x. Subtract this from the current dividend.

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

   Bring down the next term (-3).

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

7. Final division: Divide the leading term of the new dividend (3x) by the leading term of the divisor (x). This gives us 3. Write this above the division symbol.

       x^2 + 4x + 3 ____
   x - 1 | x^3 + 3x^2 - x - 3 
           x^3 - x^2
           -------
               4x^2 - x
               4x^2 - 4x
               -------
                     3x - 3
                     3x - 3
                     -------
                         0
   

   Multiply the new quotient (3) by the divisor (x - 1) to get 3x - 3. Subtract this from the current dividend, leaving a remainder of 0.

Conclusion

The result of the long division is:

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

This means that (x - 1) is a factor of the polynomial (x^3 + 3x^2 - x - 3).