When a polynomial p(x) is divided by (2x + 1), is it possible to have (x − 1) as a remainder? Justify your answer.
When we divide a polynomial by another polynomial of degree, say 'm', then the remainder left is always of a degree less than m, i.e. the remainder can be at most of degree '(m – 1)'.
Here, if we divide p(x) by (2x + 1), i.e. we divide p(x) by a polynomial of degree 1, then the remainder can be of degree '0', i.e. a constant polynomial.
Since degree of (x – 1) is 1, which is greater than '0', it is not possible to have (x – 1) as a remainder when p(x) is divided by (2x + 1).
RS Aggarwal (2015)