Division of polynomials

Division of polynomials. What means to divide one polynomial  P  by another Q ? It means to find polynomials M  ( quotient ) and N  ( remainder ), satisfying the two requirements:

          1).  An equality  MQ + N = P   takes place;
          2).  A degree of polynomial  N  is less than a degree of polynomial Q . 

Division of polynomials can be done by the following scheme ( long division ):

1)  Divide the first term 16a³ of the dividend by the first term  4a² of the divisor;  the result  4a  is the first term of the quotient.

2)  Multiply the received term 4a  by the divisor  4a² – a + 2; write the result  16a³ – 4a² + 8a  under the dividend, one similar      term under another.

3)  Subtract terms of the result from the corresponding terms of the dividend and move down the next by the order term 7 of the     dividend; the remainder is 12a² – 13a + 7 .

4)  Divide the first term 12a² of this expression by the first term 4a² of the divisor;  the result 3 is the second term of the quotient.

5)  Multiply the received second term 3 by the divisor 4a² – a + 2; write the result 12a² – 3a + 6 again under the dividend, one      similar term under another.

6)  Subtract terms of the result from the corresponding terms of the previous remainder and receive the second remainder:
     – 10a + 1. Its degree is less than the divisor degree,  therefore the division has been finished. The quotient is  4a + 3,
     the remainder is  – 10a + 1.

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