Upper and Lower Bounds
Upper and Lower Bounds - How it Works - Video
First Theorem on Bounds
First Theorem on Bounds:
We can use this theorem to narrow down our search for the real zeros by solving for the upper and lower bounds.
Statement 1 states if the third row or bottom row in synthetic division is all positives or zero. We have found an upper bound. If you choose the next smallest number, and it is not all positives or zeros then you have found the least upper bound.
Statement 2 states if the third row or bottom row in synthetic division is alternates positive and negative or zero. We have found a lower bound. If you choose the next biggest number, and it does not alternate then you have found the greatest lower bound.
Example 1
Example 1:
We could just pick any numbers to start, but we can divide the factors of the constant by the factors of the leading coefficient to narrow down our search.
Our constant is 8, and the factors of 8 are 1, 2, 4, and 8, but we have to use the ± of each one. Our leading coefficient is 2, and the factors of 2 are 1 and 2, but we have to use the ± of each one. Let's start with the biggest one, 8. There is no magic trick here. You will have to do brute force here and keep going until you find the correct number for the bounds, but at least we know the numbers are between -8 and 8.
After completing the synthetic division for 8, we find that the numbers in the third row are all positive or zeros. That means that 8 is an upper bound. If the question only asks for an upper bound and not the least, you are good, but let's continue since our numbers are quite high.
The next number that we will try is 4. Once again after completing the synthetic division for 4, we find that the numbers in the third row are all positive or zeros. 4 is the least upper bound because after completing the synthetic division for 3, the numbers in the third row are not all positive or zeros.
Now that we have our least upper bound. Let's find our greatest lower bound.
After completing the synthetic division for -1, we find that the numbers are not all positive (or zeros) and do not alternate. So -1 is not an upper bound nor a lower bound.
After completing the synthetic division for -2, we find that the numbers in the third row alternate positive and negative (remember zeros can positive or negative in this scenario). -2 is the greatest lower bound because -1 does not alternate positive and negative.
Let's do one more just to clear things up. After completing the synthetic division for -3, we find that the numbers in the third row alternate positive and negative (remember zeros can positive or negative in this scenario). That means that -3 is a lower bound as well.
Now we have the smallest interval that our real zeros could be. That interval is [-2, 4].
The interval could be bigger, but most of the time, the smallest interval is required.
Now let's take a look a graph so we can see visually how this helps. There are two dotted lines to represent the upper and lower bounds of our real zeros. We have three real zeros at -1.45, 0.90, and 3.05. Each of those numbers fall in the interval.
Second Theorem on Bounds
Second Theorem on Bounds:
We can use this theorem to find the window on our calculator. Since we are just looking at the zeros, the extrema or the turning points do not matter. So we can zoom in on just the x-axis.
Example 1 with Theorem 2
Example 1 with Theorem 2:
Here we can use the second theorem on upper bounds to graph. When we try to find the zeros, the turns or having the extrema does not matter. So on the graph, we can't see the entirety of it, but that is okay since we just want to look at the zeros.
To find that window, we take the absolute of the largest coefficient divide that by the absolute of the leading coefficient and 1 to the result of that.
In this case, we have |-6| / |2| + 1 => 3 + 1 => 4. So our window is the ± of the result, (-4, 4).