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Polynomial long division with remainder

Polynomial long division is simple.

The following example will help you to learn how to do the polynomial long division with a remainder.

We will explain how to divide a polynomial by a polynomial step by step.

Polynomial long division with remainder

Simple example.

Our problem is:

(8a2 - 10a - 5)/(2a + 1)

In the example the 8a2 - 10a - 5 is a dividend and the 2a + 1 is a divisor.

We will use the long division symbol:

                    
2a + 1)8a2 - 10a - 5

Divide the first term of the dividend (the 8a2) by the first term of the divisor (the 2a):

8a2/2a = 4a

Put the result 4a over the long division symbol:

      4a            
2a + 1)8a2 - 10a - 5

Multiply the 4a through the divisor 2a + 1:

4a(2a + 1) = 8a2 + 4a

Subtract the 8a2 + 4a from the two leading terms of the dividend:

      4a            
2a + 1)8a2 - 10a - 5
     -(8a2 + 4a)
            -6a

Carry down the last dividend term:

      4a            
2a + 1)8a2 - 10a - 5
     -(8a2 + 4a)
            -6a - 5

Divide the first term of the -6a - 5 (the -6a) by the first term of the divisor (the 2a):

-6a/2a = -3

Put the result -3 over the long division symbol:

      4a - 3        
2a + 1)8a2 - 10a - 5
     -(8a2 + 4a)
            -6a - 5

Multiply the -3 through the divisor 2a + 1:

-3(2a + 1) = -6a - 3

Subtract the -6a - 3 from the -6a - 5:

      4a - 3        
2a + 1)8a2 - 10a - 5
     -(8a2 + 4a)
            -6a - 5
          -(-6a - 3)
               -2

This -2 is a remainder.

The final result of the long division with the remainder:

(8a2 - 10a - 5)/(2a + 1) = 4a - 3 - 2/(2a + 1)