Given the polynomial function: $$f(x) = 2x^3 + 3x^2 - 29x - 60.$$ (a) Find the remainder when $f(x)$ is divided by $(x + 2)$ - Edexcel - A-Level Maths Pure - Question 6 - 2006 - Paper 2

To apply the factor theorem, we evaluate $f(-3)$:

$f(-3) = 2(-3)^3 + 3(-3)^2 - 29(-3) - 60$

Calculating this:

$f(-3) = 2(-27) + 3(9) + 87 - 60$
$= -54 + 27 + 87 - 60 = 0.$

Since $f(-3) = 0$, it follows that $(x + 3)$ is indeed a factor of $f(x)$.

Now that we know that $(x + 3)$ is a factor, we can factor $f(x)$ as follows:

$f(x) = (x + 3)(2x^2 + ax + b).$

To find the coefficients $a$ and $b$, we can perform polynomial long division or synthetic division to divide $f(x)$ by $(x + 3)$, leading to:

1. Divide $f(x)$ by $(x + 3)$, giving us a quadratic $2x^2 + 3x - 20$.
2. Next, we factor the quadratic:

$2x^2 + 3x - 20 = (2x - 5)(x + 4).$

Putting it all together, we have:

$f(x) = (x + 3)(2x - 5)(x + 4).$