Now that we have our intervals, we can go ahead and determine the behavior of our polynomial function between these intervals. The way we can go about doing this is to choose a number between each of our intervals and plug that number into our polynomial function to see if we get a positive or negative answer.
Since we have four intervals, we will pick four numbers to plug in and evaluate, one for each interval. Our first interval is between negative infinity and -1. We can pick -2 since that number is inside this interval. Plugging in -2 into our polynomial, we get f(-2) = (-2 - 7)(-2 + 1)(-2 - 2) = (-9)(-1)(-4) = -36. Our answer is negative, so our function is negative in the first interval.
Let's look at the second interval. Our interval is between -1 and 2. We can pick an easy number like 0 inside this interval. Plugging in 0 into our polynomial function, we get f(0) = (0 - 7)(0 + 1)(0 - 2) = (-7)(1)(-2) = 14. Our answer is positive, so our function is positive in this second interval.
What about the third interval? Which number would you pick? I'm going to pick 3 because 3 is a smaller number, and I find it easier to work with smaller numbers. Plugging in 3, we get f(3) = (3 - 7)(3 + 1)(3 - 2) = (-4)(4)(1) = -16. We have a negative answer, so this third interval is negative.
For the last interval, I'm going to pick the number 10 because, of the larger digits, I find 10 easier to work with. You can pick whatever number you find easy to work with. Plugging in 10, we get f(10) = (10 - 7)(10 + 1)(10 - 2) = (3)(11)(8) = 264. Our answer is positive, so our function is a positive in this interval. We can make a table of this information:
| Interval |
Function Behavior |
| (negative infinity, -1) |
- |
| (-1, 2) |
+ |
| (2, 7) |
- |
| (7, positive infinity) |
+ |
From this information, we see that our function first rises and passes through our first solution, then dips passing through the second solution, then rises again to pass through the third and final solution. We see that the x-axis cuts our function into four distinct pieces, one for each interval, which we can see more clearly if we graph out our function:
Graph of example function
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