Describe the graph of the function f (x) = x^3 - 18x^2 + 101x - 180 Include y - intercept, x - intercept, and the shape of the graph

Let's first calculate the roots of f (x) = x³-18x²+101x-180

a) f (4) = 4³ - 18. (4²) + 101. (4) - 180 = 0, then x=4 is a root.

b) to find the 2 other roots divide x³-18x²+101x-180 by (x-4) = x²-14x+45

c) this quadratic equation has the following roots: x=5 and x=9. Hence:

f (x) = x³-18x²+101x-180 = (x-4) (x-5) (x-9)

the x-intercepts are x₁ = 4, x₂ = 5, x₃ = 9.

To find the y intercept plug x = 0 in f (x) = x³-18x²+101x-180

f (x) = y = - 180 (y intercept

The graph starts from - ∞ to + ∞ and passes through one maximum and one minimum