Synthetic Division: Finding the Quotient and Remainder of (-xﾲ + x - 5) / (x + 1)

Synthetic Division: Dividing -xﾲ + x - 5 by x + 1

Synthetic division is a shortcut method to divide polynomials, particularly useful when the divisor is a linear expression like x + 1. Here's how to find the quotient and remainder of (-xﾲ + x - 5) / (x + 1) using synthetic division:

1. Set up the Division:

• Write the coefficients of the dividend (-xﾲ + x - 5) in a row: -1 1 -5
• Write the opposite of the constant term of the divisor (x + 1) to the left: -1

This setup looks like this:

      -1 | -1   1   -5 

2. Bring Down the First Coefficient:

Bring down the first coefficient (-1) below the line.

      -1 | -1   1   -5 
          | -1  
          |--------

3. Multiply and Add:

• Multiply the divisor (-1) by the number you just brought down (-1): (-1) * (-1) = 1
• Write the result (1) in the next column.
• Add the numbers in that column: 1 + 1 = 2

      -1 | -1   1   -5 
          | -1   1
          |--------
            -1   2  

4. Repeat the Process:

• Multiply the divisor (-1) by the new number below the line (2): (-1) * 2 = -2
• Write the result (-2) in the next column.
• Add the numbers in that column: -2 + (-5) = -7

      -1 | -1   1   -5 
          | -1   1   -2
          |--------
            -1   2   -7 

5. Interpret the Results:

• The numbers below the line (-1, 2) are the coefficients of the quotient, starting with one degree less than the dividend. In this case, the quotient is -x + 2.
• The last number below the line (-7) is the remainder.

Therefore, (-xﾲ + x - 5) / (x + 1) = -x + 2 - 7/(x + 1).