(x^4)/(x-1) Long Division

Jasonbradley

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Jun 17, 2024

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Understanding Long Division with (x^4)/(x-1)

Long division is a fundamental process in algebra, often used to simplify rational expressions. Let's explore how to perform long division with the expression (x^4)/(x-1).

Setting up the Division

1. Write the dividend and divisor:

   • Dividend: x^4 (notice that we can write this as x^4 + 0x^3 + 0x^2 + 0x + 0)
   • Divisor: x - 1

2. Set up the division structure:

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

Performing the Division

1. Focus on the leading terms: Divide the leading term of the dividend (x^4) by the leading term of the divisor (x). This gives us x^3.

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

2. Multiply the quotient by the divisor: Multiply x^3 by (x - 1) to get x^4 - x^3.

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

3. Subtract the result: Subtract the result from the dividend. Notice that the x^4 terms cancel out.

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

4. Bring down the next term: Bring down the next term (0x^2) from the dividend.

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

5. Repeat steps 1-4: Divide the leading term of the new dividend (x^3) by the leading term of the divisor (x) to get x^2.

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

6. Continue the process: Continue bringing down terms and repeating the process until the degree of the remaining dividend is less than the degree of the divisor.

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

The Result

The result of the long division is:

• Quotient: x^3 + x^2 + x + 1
• Remainder: 1

Therefore, we can write the expression (x^4)/(x-1) as:

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

This illustrates how long division allows us to simplify rational expressions and express them as a combination of a polynomial and a fractional remainder.