f(x) = 3x^3 - 5x^2 - 58x + 40 (a) Find the remainder when f(x) is divided by (x - 3) - Edexcel - A-Level Maths Pure - Question 4 - 2010 - Paper 3

Since (x - 5) is given as a factor of f(x), we can factor f(x) as follows:

Let f(x) = (x - 5)(Ax^2 + Bx + C) for some constants A, B, and C. Using polynomial long division on f(x):

1. Divide f(x) by (x - 5) to find the quadratic.

Perform the polynomial long division, which yields:

$$f(x) = (x - 5)(3x^2 + 10x - 8).$$

2. Now, set the factored form equal to zero:

$$(x - 5)(3x^2 + 10x - 8) = 0.$$

This gives us the first solution: x = 5.

3. Next, solve the quadratic equation $3x^2 + 10x - 8 = 0$ using the quadratic formula:

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

where a = 3, b = 10, and c = -8:

$$x = \frac{-10 \pm \sqrt{10^2 - 4(3)(-8)}}{2(3)} = \frac{-10 \pm \sqrt{100 + 96}}{6} = \frac{-10 \pm \sqrt{196}}{6} = \frac{-10 \pm 14}{6}.$$

4. This gives us the solutions:

$$x_1 = \frac{4}{6} = \frac{2}{3}$$

and

$$x_2 = \frac{-24}{6} = -4.$$

Thus, the complete set of solutions to f(x) = 0 is x = 5, x = \frac{2}{3}, and x = -4.