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EXERCISE 2.3
1. Find the remainder when x + 3x2 + 3x + 1 is divided by long division method
1
(i) x + 1
(ii) x -
(iii) *
2
(iv) x + 7
(v) 5 + 2x
2. Find the remainder when x - ar? + 6x-a is divided by x -- a.
3. Check whether 7 + 3x is a factor of 3x + 7x.
2.5 Factorisation of Polynomials
Let us now look at the situation of Example 10 above more closely. It tells us that since
the remainder, 9
C)
0, (2t + 1) is a factor of q(t), i.e., q(t) = (2t + 1) g(t) for some
polynomial g(t). This is a particular case of the following theorem.
Factor Theorem: If p(x) is a polynomial of degree n 21 and a is any real number, then
(i) x - a is a factor of p(x), if p(a) = 0, and
(ii) p(a) = 0, if x - a is a factor of p(x).
This actually follows immediately from the Remainder Theorem, but we shall not prove
it here. However, we shall apply it quite a bit, as in the following examples.
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