Polynomial long division

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

Polynomial long division

Explaining in more details the polynomial long division in the figure above

The polynomial long division above is a summary showing how to do long division with polynomials. Read example #1 to fully understand it!

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

Another straightforward example showing polynomial long division

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

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