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
2). A degree of polynomial N is less than a degree of polynomial Q .
Division of polynomials can be done by the following scheme (
1) Divide the first term 16
of the dividend by the first term 4
of the divisor; the result
is the first term of the quotient.
2) Multiply the received term 4 a by the divisor 4 a 2 – a + 2; write the result 16 a 3 – 4 a 2 + 8 a 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 12 a 2 – 13 a + 7 .
4) Divide the first term 12 a 2 of this expression by the first term 4 a 2 of the divisor; the result 3 is the second term of the quotient.
5) Multiply the received second term 3 by the divisor 4 a 2 – a + 2; write the result 12 a 2 – 3 a + 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:
– 10 a + 1. Its degree is less than the divisor degree, therefore the division has been finished. The quotient is 4 a + 3,
the remainder is – 10 a + 1.