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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 16
a
^{3}
of the dividend by the first term 4
a
^{2}
of the divisor; the result 4
a
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.
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