GCSE or A-level Maths: How can I find the x and y intercepts of a cubic function?

Assume we have the function:
y = x ( x² + 8x - 9)
This function is known as a cubic function as when multiplying out the brackets, the highest function of x would be x³.Note: If the highest order of x was x² then this would be called a quadratic.
To find the x intercepts means to find when this curve, when graphed on a y/x axis, would cross the x-axis. This is also known to be when y is equal to 0.
To find the y intercept means to find when this curve, when graphed on a y/x axis, would cross the y-axis. This is also known to be when x is equal to 0. Note: for the y intercept, notice how I have used the word intercept and not intercepts. This is because for cubics, the curve can only cross the y-axis once and only once due to its nature.
Lets start off by trying to find the x intercepts. To do this, we will equate y = 0 into the function:
y = x ( x² + 8x - 9) 0 = x ( x² + 8x - 9)
To determine the intercepts, each indivicual value of x will have its own bracket spereate bracket. For example, review the following functions and their equivilant x intercepts:
y = (x + 3)(x - 2), this QUADRATIC function has x intercepts of: x = -3 and x = 2y = x (x + 1), this QUADRATIC function has x intercepts of: x = 0 and x = -1y = (x + 3)(x - 2)(3x + 4), this CUBIC function has x intercepts of: x = -3, x = 2 and x = -4/3
From the above examples, you should be able to notice a pattern. How this actually works is simple. Lets work through this in our example. We currently have:
0 = x ( x² + 8x - 9)
Lets start of by factorising the brackets. (Assuming factorising is covered):
0 = x (x + 9)(x - 1)
Now, lets work with the first x. of we divide both sides of this equation by (x + 9)(x - 1), the remainder of the equation, this gives us:
0 = x
Now we know one of the x intercepts. Now lets look at the second x, lets divide both sides of this equation by x (x - 1):
x + 9 = 0
Therefore x = -9, our second intercept. And finally the last x, by dividing through both sides by x (x + 9):
x - 1 = 0
Thus, x = 1, our third and final intercept of the x-axis. Now, to find the y intercept, simply return to any form of the origional function, factorised or non factorised. Equate all x's to zero as we want the y intercept, to get the following:
y = x ( x² + 8x - 9)y = 0 ( 0² + 0x - 9)y = 0 ( -9) = 0
Therefore, the curve intercepts the y-axis at y = 0.