We found that x4−2x3−2x2−2x−3 is a quartic function with -1 and 3 as its only real zeros. Note that, since there are infinite quartic functions with only the two given real zeros, this is just an example solution.
Alternative Solution
Solving the exercise by using transformations of a graph.
If you like graphs, there is another way of solving this exercise. Consider the parent functiony=x4.
Since the difference between the given zeros is four, we have to find a quartic function where the roots are 4 units apart, and having them centered at 0 is the easiest. Then we can use translations to move the graph over to the correct place. Let's find a quartic with zeros at ±2, because that will add up to 4.
To do so, we need to translate the parent function (2)4=16 units down. Therefore, we need to consider the function y=x4−16.
Finally, -1 and 3 are 1 unit to the right of -2 and 2, respectively.
This means that we need to translate the last function 1 unit to the right. By doing this, we can obtain the function y=(x−1)4−16.
We can see that y=(x−1)4−16 is a quartic function with -1 and 3 as its only real zeros. Note that, although this function is different than the one we found using the algebraic method, it is also a correct answer.
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