f(x) = 2x^3 + x^2 - 5x + c, where c is a constant - Edexcel - A-Level Maths Pure - Question 3 - 2006 - Paper 2

Given:

$$f(x) = 2x^3 + x^2 - 5x + 2\$$

To factorise, we can first factor out common terms using synthetic division, dividing by (x - 1):

Upon performing synthetic division, we find:

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

Now we need to factor the quadratic expression:

$$2x^2 + 3x - 2 = (2x - 1)(x + 2)\$$

Thus, the complete factorization is:

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

To find the remainder when dividing f(x) by (2x - 3), we can use the Remainder Theorem. We first need to find the value of x where 2x - 3 = 0:

$$2x = 3 \\ x = \frac{3}{2} \\$$

Now we substitute this value into f(x):

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

Calculating this:

$$f(\frac{3}{2}) = 2\frac{27}{8} + \frac{9}{4} - \frac{15}{2} + 2 = \frac{27}{8} + \frac{18}{8} - \frac{60}{8} + \frac{16}{8} = \frac{-9}{8} + \frac{-6}{8} = -\frac{15}{8} \\$$

Thus, the remainder is:

$-\frac{15}{8}$