Question

How do I find real zeros of `f(x) = x^5 - 3x^4 + 11x - 9` on a TI-84?

Answer 1

The zeros are 1, 1.4767, and 2.5490.

The most difficult part of finding zeros on a graphing calculator is setting a proper window. This is easier if you know the properties of polynomial functions. The first property is that the maximum number of real roots is the order of the polynomial (or the highest degree); in the case of your question, it is 5.

The second property is the factor property which states that any factor of the polynomial, must be a factor of the constant term; in the case of your question, it is 9. This means the window should be set to `x in [-9, 9]`. Since we don't care about minimums or maximums when looking for zeros, we can set `y in [-5,5]` (if the curve is really steep, you can make the range larger). Press the "WINDOW" button on the top row to set your window size.

Next, you need to enter the function into your calculator. You do this by pressing the "Y=" button at the top left of your calculator. Then press "GRAPH". Once you see the zeros, you may want to adjust your window size to see the zero more clearly; in the case of your question `x in [0,3]`.

Now, you can press the "2nd" key followed by the "TRACE" key to perform "CALC". You will want to press "2" to compute zeros. When selecting your left and right bounds, you want to make sure that your curve is strictly increasing or decreasing. If there is a hill or valley in your bounds, you will most likely get an error. Your guess can simply be the right bound. Repeat this for as many zeros as you can see; in your case there are 3.