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

Jasonbradley

5 min read

·

Jun 17, 2024

20

Share

Simplifying the Expression (x^3 + 3x^2 - x - 3) / (x - 1)

This expression represents a rational function, a fraction where both the numerator and denominator are polynomials. To simplify this expression, we can use polynomial long division.

Performing Long Division

1. Set up the division:

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

2. Divide the leading terms:

   • The leading term of the divisor (x - 1) is 'x'.
   • The leading term of the dividend (x^3 + 3x^2 - x - 3) is 'x^3'.
   • x^3 / x = x^2
   • Write x^2 above the dividend:

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

3. Multiply the divisor by the quotient:

   • (x - 1) * x^2 = x^3 - x^2
   • Write the result below the dividend:

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

4. Subtract the result from the dividend:

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

5. Bring down the next term:

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

6. Repeat steps 2-5:

   • Divide the leading term of the new dividend (4x^2) by the leading term of the divisor (x): 4x^2 / x = 4x
   • Write 4x above the dividend:

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

   • Multiply the divisor (x - 1) by 4x: (x - 1) * 4x = 4x^2 - 4x
   • Write the result below the dividend:

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

   • Subtract the result:

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

   • Bring down the next term:

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

7. Repeat steps 2-5 again:

   • Divide the leading term of the new dividend (3x) by the leading term of the divisor (x): 3x / x = 3
   • Write 3 above the dividend:

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

   • Multiply the divisor (x - 1) by 3: (x - 1) * 3 = 3x - 3
   • Write the result below the dividend:

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

   • Subtract the result:

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

8. The remainder is 0: This means that the divisor (x - 1) divides evenly into the dividend (x^3 + 3x^2 - x - 3).

Simplified Expression

The result of the long division shows that: (x^3 + 3x^2 - x - 3) / (x - 1) = x^2 + 4x + 3

This simplified expression is a quadratic polynomial.