# Write a polynomial function in standard form with zeros

Example 1 Find the zeroes of each of the following polynomials. So, before we get into that we need to get some ideas out of the way regarding zeroes of polynomials that will help us in that process.

Putting this information together with the Leading Coefficient Test we can determine the end behavior of the graph of our given polynomial: Since the degree of the polynomial, 3, is odd and the leading coefficient, 5, is positive, then the graph of the given polynomial falls to the left and rises to the right.

Example 1: Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial. If you said -1, pat yourself on the back!! Again, if we go back to the previous example we can see that this is verified with the polynomials listed there.

Example 2 List the multiplicities of the zeroes of each of the following polynomials.

## How to find the zeros of a function

We can go back to the previous example and verify that this fact is true for the polynomials listed there. If you said 5, you are right on!! In this section we have worked with polynomials that only have real zeroes but do not let that lead you to the idea that this theorem will only apply to real zeroes. The factor theorem leads to the following fact. This example leads us to several nice facts about polynomials. In each case we will simply write down the previously found zeroes and then go back to the factored form of the polynomial, look at the exponent on each term and give the multiplicity. It is completely possible that complex zeroes will show up in the list of zeroes. If you said 4, you are right on!! The zero factor property can be extended out to as many terms as we need. To do this all we need to do is a quick synthetic division as follows.

Note as well that some of the zeroes may be complex. If you said -7, pat yourself on the back!!

### How to find the equation of a polynomial function given points

If you said 5, you are right on!! Note as well that some of the zeroes may be complex. If you said 6, you are right on!! Again, if we go back to the previous example we can see that this is verified with the polynomials listed there. This fact is easy enough to verify directly. If you said 3, you are right on!! Another way to say this fact is that the multiplicity of all the zeroes must add to the degree of the polynomial. To do this all we need to do is a quick synthetic division as follows. Do not worry about factoring anything like this.

Zeroes with a multiplicity of 1 are often called simple zeroes. This example leads us to several nice facts about polynomials.

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