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At this point, you discover that you cannot divide 32 by 41 and produce a whole
number. For this reason, you place a 0 in the 1's place. To indicate that the
division has resulted in a remainder, you show a fraction of 32/41.
Long Division in Algebra
When you perform divisions involving polynomials, you use an approach similar to
the one you employ for arithmetic. To perform such divisions, you group like terms
and combine them. You also order the terms relative to their degree in descending
order. Likewise, it is helpful if you remove fractions and in other respects simplify
the expressions as much as possible before attempting the division.
As an example of a polynomial division problem, consider this expression:
2 x 2
þ 3 x þ 7
x þ 3
The terms in the numerator appear in descending order with respect to the degrees
of their exponents. In this case, the largest degree is that of the square of x .The
same applies to the terms of the denominator, where the highest degree is 1.
To set up the division, you proceed in the same manner you did when you
performed long division involving whole numbers:
To perform the division, you concentrate first on the term with the exponent of
the highest degree. Carrying out such a division is analogous to long division in
arithmetic. You can envision 8642 in this way:
þ 2 10 0
In the previous section, for example, in essence you first divide 8 10 3 by 41. Ten
raised to the power of 3 constituted the highest degree of ten as given in the problem.
Having dealt with the highest degree of 10, you then move on to the second highest
degree.
8 10 3
þ 6 10 2
þ 4 10 1
In this instance, you concentrate first on the highest power of x , which is 2.
Isolating the terms involved, the division takes this form:
2 x 2
x ¼ 2 x 2 1
¼ 2 x
 
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