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 4 - 2011 - Paper 2

To use the factor theorem to show that $(x+1)$ is a factor of $f(x)$, we need to evaluate $f(-1)$:

$f(-1) = 2(-1)^3 - 7(-1)^2 - 5(-1) + 4$ $= 2(-1) - 7(1) + 5 + 4$ $= -2 - 7 + 5 + 4$ $= 0$

Since $f(-1) = 0$, by the factor theorem, $(x+1)$ is indeed a factor of $f(x)$.

To factorise $f(x)$ completely, we start with the factor found in part (b), $(x+1)$. We can perform polynomial long division of $f(x)$ by $(x+1)$:

1. Perform the division:

   • Divide $2x^3$ by $x$ to get $2x^2$.
   • Multiply $(x+1)$ by $2x^2$ gives $2x^3 + 2x^2$.
   • Subtract: $f(x) - (2x^3 + 2x^2) = -9x^2 - 5x + 4$.

2. Repeat the process:

   • Divide $-9x^2$ by $x$ to get $-9x$.
   • Multiply: $-9x(x + 1) = -9x^2 - 9x$.
   • Subtract: $(-9x^2 - 5x + 4) - (-9x^2 - 9x) = 4x + 4$.

3. Divide $4x + 4$ by $(x + 1)$:

   • Divide $4x$ by $x$ to get $4$.
   • Multiply: $4(x + 1) = 4x + 4$.
   • Subtract: $0$.

Thus, $f(x) = (x + 1)(2x^2 - 9x + 4)$. Now factor this quadratic: Using the quadratic formula: