Polynomial long division

Learn polynomial long division with these two examples that are easy to follow and straight to the point.

Example #1

Divide x² + 3x - 10 by x - 2

Divide the leading term of x² + 3x - 10 by the leading term of x - 2

x² ÷ x = x

Write x as the first term of the quotient 

          x                         
 x - 2)  x² + 3x - 10

Multiply the first term of the quotient by the divisor

x(x - 2) = x² - 2x

Subtract x² - 2x from the dividend

          x                         
 x - 2)  x² + 3x - 10
          -( x² - 2x)

          x                         
 x - 2)  x² + 3x - 10
          -x² + 2x
       ______________
                   5x  

Bring down -10

          x                         
 x - 2)  x² + 3x - 10
          -x² + 2x
       ______________
                   5x  - 10

Divide the leading term of 5x - 10 by the leading term of x - 2

5x ÷ x = 5

Write 5 as the second term of the quotient. Since 5 is positive, you can put a + sign between the first term and the second term.

            x  + 5                     
 x - 2)  x² + 3x - 10
          -x² + 2x
       ______________
                   5x  - 10

Multiply the second term of the quotient by the divisor

5(x - 2) = 5x - 10

Subtract 5x - 10 from 5x - 10

          x  + 5                     
 x - 2)  x² + 3x - 10
          -x² + 2x
        ______________
                   5x  - 10
                  -5x + 10
              __________
                       0       

(x² + 3x - 10) ÷ (x - 2) = x + 5

Example #2

Divide x² - 5x + 1 by x + 3

Divide the leading term of x² - 5x + 1 by the leading term of x + 3

x² ÷ x = x

Write x as the first term of the quotient 

          x                         
 x + 3)  x² - 5x + 1

Multiply the first term of the quotient by the divisor

x(x + 3) = x² + 3x

Subtract x² + 3x from the dividend

          x                         
 x + 3)  x² - 5x + 1
          -( x² + 3x)

          x                         
 x + 3)  x² - 5x + 1
          -x² - 3x
       ______________
                -8x  

Bring down 1

          x                         
 x + 3)  x² - 5x + 1
          -x² - 3x
       ______________
                   -8x  + 1

Divide the leading term of -8x  + 1 by the leading term of x + 3

-8x ÷ x = -8

Write -8 as the second term of the quotient. Notice that putting a + sign between the first term and the second term does not change the problem.

            x + -8                     
 x + 3)  x² - 5x + 1
          -x² - 3x
       ______________
                   -8x  + 1

Multiply the second term of the quotient by the divisor

-8(x + 3) = -8x - 24

Subtract -8x - 24 from -8x + 1

          x  + -8                     
 x + 3)  x² - 5x + 1
         -x² - 3x
        ______________
                   -8x  + 1
                    8x + 24
              __________
                           25       

(x² - 5x + 1) ÷ (x + 3) = x + -8 with a remainder of 25