Given the polynomial function: $$f(x) = 2x^3 - 7x^2 - 5x + 4$$ (a) Find the remainder when $f(x)$ is divided by $(x-1)$ - Edexcel - A-Level Maths Pure - Question 3 - 2011 - Paper 2

According to the Factor Theorem, $(x-c)$ is a factor of $f(x)$ if $f(c) = 0$. In this case, we need to evaluate $f(-1)$:

$f(-1) = 2(-1)^3 - 7(-1)^2 - 5(-1) + 4$

Calculating:

$= 2(-1) - 7(1) + 5 + 4$

$= -2 - 7 + 5 + 4 = 0$

Since $f(-1) = 0$, this confirms that $(x+1)$ is indeed a factor of $f(x)$.

To factor $f(x)$ completely, we already have one factor $(x+1)$. We can perform polynomial long division or synthetic division of $f(x)$ by $(x+1)$:

$f(x) = (x+1)(2x^2 - 9x + 4)$

Now we need to factor $2x^2 - 9x + 4$. To factor this quadratic, we look for two numbers that multiply to $2*4 = 8$ and add to $-9$. The correct factors are $-8$ and $-1$. Therefore, we can factor further:

$2x^2 - 9x + 4 = 2x^2 - 8x - x + 4 = 2x(x - 4) - 1(x - 4) = (2x-1)(x-4)$

Putting it all together:

$f(x) = (x+1)(2x-1)(x-4)$

This is the complete factorization of $f(x)$.