We are applying here an algorithm called Euclid’s division algorithm.
This says that
If p(x) and g(x) are any two polynomials with g(x) ¹ 0, then we can find polynomials
q(x) and r(x) such that
p(x) = g(x) × q(x) + r(x),
where either r(x) = 0 or degree of r(x) < degree of g(x) if r(x) ¹ 0
This result is known as the Division Algorithm for polynomials.
Now, we have the following results from the above discussions
(i) If g(x) is a linear polynomial then r(x) = r is a constant.
(ii) If degree of g(x) = 1, then degree of p(x) = 1 + degree of q(x)0
(iii) If p(x) is divided by (x – a), then the remainder is p(a)0
(iv) If r = 0, we say q(x) divides p(x) exactly or q(x) is a factor of p(x).
Let us now take some examples to illustrate its use.
Example-10.
Divide 3x
2
– x
3
– 3x + 5 by x – 1 – x
2
, and verify the division algorithm.
Solution :
Note that the given polynomials are not in standard form. To carry out division, we
first write both the dividend and divisor in decreasing orders of their degrees.
So, dividend = – x
3
+ 3x
2
– 3x + 5 and
divisor = – x
2
+ x – 1.
Division process is shown on the right side.
We stop here since degree of the remainder is
less than the degree of (–x
2
+ x – 1), the divisor.
divisor = – x
2
+ x – 1.