Question

Graph the polynomial f(x)=(x+6)(x+4)^2 (x+2)^3(x-3) (x-5)

a) Describe the interval notation the set of all x such that f(x)>0.
b) Graph on a number line the set of all x such that f(x) > 0.
c)Describe in interval notation the set of all x such that f(x) ≥ 0.
d).Graph on a number line the set of all x such that f(x) ≥ 0.

Answer 1

These interval questions are easy.
f(x) is a 8th degree polynomial with positive first coefficient, so its end behavior on both ends is that it rises up to +infinity.

So, look at the smallest and largest roots: -6 and 5

f(x) > 0 for x < -6 and x > 5

So, what about in between? The trick here is to recall that

at a root of odd order, the graph crosses the x-axis (think x or x^3)

at a root of even order, the graph just touches the x-axis and then goes back the way it came. (think x^2)

So, the graph of f(x) crosses the x-axis at x = -6,-2,3,5

It just touches the x-axis at x = -4

So, looking at the x-axis, we have

<-- +++ (-6) --- (-4) --- (-2) +++ (3) --- (5) +++ -->