Given that f(1) = 0, (a) find the value of c, (b) factorise f(x) completely, (c) find the remainder when f(x) is divided by (2x - 3). - Edexcel - A-Level Maths Pure - Question 3 - 2006 - Paper 2

The polynomial can be factored by first noting that f(1) = 0, indicating (x - 1) is a factor. Perform polynomial long division of f(x) by (x - 1):

Performing the division gives:

$f(x) = (x - 1)(2x^2 + 3x - 2).$

Next, factor the quadratic $2x^2 + 3x - 2$. This can be factored as:

$f(x) = (x - 1)(2x - 1)(x + 2).$

To find the remainder when f(x) is divided by (2x - 3), we can use the Remainder Theorem. Substitute (x = \frac{3}{2}) into f(x):

$f\left(\frac{3}{2}\right) = 2\left(\frac{3}{2}\right)^3 + \left(\frac{3}{2}\right)^2 - 5\left(\frac{3}{2}\right) + 2.$

Calculating this gives:

= \frac{27}{4} + \frac{9}{4} - \frac{30}{4} + \frac{8}{4} = \frac{14}{4} = 3.5.$$