Division of polynomials
Division of polynomials (quotient, remainder). Long division.
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 16a^{3} of the dividend by the first term 4a^{2} of the divisor; the result
4a is the first term of the quotient.
2) Multiply the received term 4a by the divisor 4a^{2} – a + 2; write the result 16a^{3}
– 4a^{2} + 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 12a^{2} – 13a + 7 .
4) Divide the first term 12a^{2} of this expression by the first term 4a^{2} of the divisor; the result 3 is the second
term of the quotient.
5) Multiply the received second term 3 by the divisor 4a^{2} – a + 2; write the result
12a^{2} – 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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