# 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 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.