College Algebra, 2013a

272 Polynomial Functions
believed. After resizing the window, we see not only the relative maximum but also a relative
minimum 2 just to the left of x = −1 which shows us, once again, that Mathematics enhances the
technology, instead of vice-versa.
Our next example shows how even a mild-mannered polynomial can cause problems.
Example 3.3.4. Let f(x) =x 4 + x 2 − 12.
1. Use Cauchy’s Bound to determine an interval in which all of the real zeros of f lie.
2. Use the Rational Zeros Theorem to determine a list of possible rational zeros of f.
3. Graph y = f(x) using your graphing calculator.
4. Find all of the real zeros of f and their multiplicities.
Solution.
1. Applying Cauchy’s Bound, we find M = 12, so all of the real zeros lie in the interval [−13, 13].
2. Applying the Rational Zeros Theorem with constant term a 0 = −12 and leading coefficient
a 4 = 1, we get the list {± 1, ± 2, ± 3, ± 4, ± 6, ± 12}.
3. Graphing y = f(x) on the interval [−13, 13] produces the graph below on the left. Zooming
in a bit gives the graph below on the right. Based on the graph, none of our rational zeros
will work. (Do you see why not?)
2 This is an example of what is called ‘hidden behavior.’