Question

Please help i’m really struggling!!!!

Answer 1

x-intercepts are points of the form (x, 0), and y-intercepts are points of the form (0, y).

You find the x-intercepts by solving for x such that y = f(x) = 0:

$x^3-9x^2+20x-12=0$

You know x - 6 is a factor of f(x), which means x = 6 is a solution to the above equation, and that means that (6, 0) is one x-intercept.

You can find the other two by dividing f(x) by x - 6 (i.e. further factorizing f(x)).

$\dfrac{x^3-9x^2+20x-12}{x-6}=x^2-3x+2=(x-2)(x-1)$

So we have

$x^3-9x^2+20x-12=(x-6)(x-2)(x-1)=0\implies x=6,x=2,x=1$

and this means the x-intercepts are (6, 0), (2, 0) and (1, 0).

The y-intercept is easier to find; set x = 0, then solve for y = f(0):

$y=0^3-9\cdot0^2+20\cdot0-12=-12$

and so the y-intercept is (0, -12).