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

Using the Factor Theorem, if (x + 1) is a factor of f(x), then f(-1) should equal zero.

Calculating f(-1):

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

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

We have established that (x + 1) is a factor. We can use polynomial long division to factor f(x).

When we divide f(x) by (x + 1), we find:

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

Next, we need to factor the quadratic $2x^2 - 9x + 4$ completely. We can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 2$, $b = -9$, and $c = 4$:

Calculating the discriminant:

$D = (-9)^2 - 4(2)(4) = 81 - 32 = 49$

Now substituting into the formula:

$x = \frac{9 \pm 7}{4}$

This gives us two roots:

$x_1 = \frac{16}{4} = 4$ $x_2 = \frac{2}{4} = \frac{1}{2}$

Thus, we can factor $(2x^2 - 9x + 4)$ as:

$2(x - 4)(x - \frac{1}{2})$

Putting it all together:

$f(x) = (x + 1)(2)(x - 4)(x - \frac{1}{2})$

This is the completely factored form of f(x).