Question

Let f(x)=(x−7)(x+6)(x−5). Find the y-intercept(s), the x-intercept(s), the values of x where f(x)>0, and the values of x where f(x)<0. Do not sketch the graph.

1. Find the y-intercept(s). List your answers as points in the form (a,b)
2.Find the x-intercept(s). List your answers as points in the form (a,b)
3. What are the values of x where f(x)>0?
4. What are the values of x where f(x)<0

Answer 1

Part 1

The y-intercept is when x=0, and
`f(0)=(0-7)(0+6)(0-5)=210`. So, the y-intercept is at (0, 210).

Part 2

The x-intercept(s) are when y=0, so:

`f(x)=0\\\\(x-7)(x+6)(x-5)=0\\\\x-7=0, x+6=0, x-5=0\\\\x=-6, 5, 7`

So, the x-intercepts are: (-6, 0), (5, 0), (7, 0)

Using our roots from Part 2, we know that:

Part 3:
`(-6, 5) \cup (7, \infty)`

Part 4:
`(-\infty, -6) \cup (5, 7)`