A
(a) (i) Show that the equation y = (x - 4)(x + 1)(x - 2) can be written as y = x ^ 3 - 5x ^ 2 + 2x + 8
To expand the equation y = (x-4)(x+1)(x-2), we can use the distributive property.
First, let's multiply (x-4) and (x+1) using the FOIL method:
(x-4)(x+1) = x*x + x*1 - 4*x - 4*1
= x^2 + x - 4x - 4
= x^2 - 3x - 4
Now, let's multiply the result by (x-2):
(x^2 - 3x - 4)(x-2) = (x^2 - 3x - 4)*x - (x^2 - 3x - 4)*2
= x^3 - 3x^2 - 4x - 2x^2 + 6x + 8
= x^3 - 5x^2 + 2x + 8
Therefore, we have shown that y = (x-4)(x+1)(x-2) can be written as y = x^3 - 5x^2 + 2x + 8.