How to Perform Polynomial Long Division: Example with xﾲ - 4x + 6 divided by x - 1

Dividing Polynomials with Long Division: A Step-by-Step Example

This tutorial explains how to perform polynomial long division, a crucial skill in algebra. We'll walk through an example, dividing the polynomial xﾲ - 4x + 6 by x - 1.

1. Set up the Long Division:

         x - 1       __________________x^2 - 4x + 6 | x^2 + 0x - 4x + 6 

2. Divide the Leading Terms:

• Focus on the highest degree terms in both the divisor and dividend: xﾲ and x. - Divide xﾲ by x, which gives us x. This is the first term of our quotient.

         x - 1       __________________x^2 - 4x + 6 | x^2 + 0x - 4x + 6             - (x^2 -  x) 
  

3. Multiply and Subtract:

• Multiply the divisor (x - 1) by the first term of the quotient (x).- Subtract the result from the dividend.

         x - 1       __________________x^2 - 4x + 6 | x^2 + 0x - 4x + 6             - (x^2 -  x)       __________________                   -3x + 6
  

4. Repeat Steps 2 and 3:

• Now our new dividend is -3x + 6.- Divide the leading term of the new dividend (-3x) by the leading term of the divisor (x), which gives us -3. This is the next term of our quotient. - Multiply the divisor (x - 1) by -3 and subtract the result from the new dividend.

         x - 1       __________________x^2 - 4x + 6 | x^2 + 0x - 4x + 6             - (x^2 -  x)       __________________                   -3x + 6                   - (-3x + 3)       __________________                          3
  

5. Determine the Quotient and Remainder:

• We are left with 3, which has a lower degree than the divisor (x - 1). This is our remainder.- The terms we obtained above the division line (x - 3) form the quotient.

Therefore:

• Quotient: x - 3- Remainder: 3

This means we can express the division as: (xﾲ - 4x + 6) / (x - 1) = x - 3 + 3/(x - 1)