8. (a) Factorise completely 9x - 4x³ (b) Sketch the curve C with equation y = 9x - 4x³ Show on your sketch the coordinates at which the curve meets the x-axis - Edexcel - A-Level Maths Pure - Question 10 - 2015 - Paper 1

To sketch the curve defined by the equation, follow these steps:

1. Find the x-intercepts: Set y = 0:

   $0 = 9x - 4x^3$

   From the factorised form, the x-intercepts are:

   • x = 0
   • x = \frac{3}{2} ext{ and } x = -\frac{3}{2} \text{ from } 3 - 2x = 0 ext{ and } 3 + 2x = 0.

2. Plot key points: Find additional points by substituting values of x into the equation:

   • For example, at x = -1:

     $y = 9(-1) - 4(-1)^3 = -9 + 4 = -5$

   • At x = 1:

     $y = 9(1) - 4(1)^3 = 9 - 4 = 5$

3. Sketch the curve: Plot these points and connect them smoothly, ensuring to show the intercepts and the general shape of the polynomial.

4. Show coordinates where it meets the x-axis: Mark points where y = 0 on the x-axis.

To find the length of segment AB:

1. Determine coordinates: Use the equation y = 9x - 4x³ to find the y-coordinates:

   • For Point A at x = -2:

     $y_A = 9(-2) - 4(-2)^3 = -18 + 32 = 14$

   • For Point B at x = 1:

     $y_B = 9(1) - 4(1)^3 = 9 - 4 = 5$

   This gives us coordinates: A(-2, 14) and B(1, 5).

2. Calculate the distance AB: Use the distance formula:

   $AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}$

   Substituting the coordinates:

   $AB = \sqrt{(1 - (-2))^2 + (5 - 14)^2}$

   $= \sqrt{(1 + 2)^2 + (-9)^2}$

   $= \sqrt{3^2 + 81} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10}$

   Thus, we have length AB expressed as:

   $AB = k\sqrt{10} ext{ where } k = 3$.