# Writing a polynomial function from its zeros

We solved each of these by first factoring the polynomial and then using the zero factor property on the factored form.

This fact is easy enough to verify directly. Find the polynomial of least degree containing all of the factors found in the previous step. Find the size of squares that should be cut out to maximize the volume enclosed by the box.

Even then, finding where extrema occur can still be algebraically challenging. How To: Given a graph of a polynomial function, write a formula for the function Identify the x-intercepts of the graph to find the factors of the polynomial.

Another way to say this fact is that the multiplicity of all the zeroes must add to the degree of the polynomial.

### How to write a polynomial function from a graph

Do not worry about factoring anything like this. How To: Given a graph of a polynomial function, write a formula for the function Identify the x-intercepts of the graph to find the factors of the polynomial. So, if we could factor higher degree polynomials we could then solve these as well. 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. For now, we will estimate the locations of turning points using technology to generate a graph. Only polynomial functions of even degree have a global minimum or maximum. We will also use these in a later example. We solved each of these by first factoring the polynomial and then using the zero factor property on the factored form. Because a height of 0 cm is not reasonable, we consider only the zeros 10 and 7. Example: Using Local Extrema to Solve Applications An open-top box is to be constructed by cutting out squares from each corner of a 14 cm by 20 cm sheet of plastic then folding up the sides.

So, why go on about this? When we first looked at the zero factor property we saw that it said that if the product of two terms was zero then one of the terms had to be zero to start off with. This is a great check of our synthetic division.

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

Licenses and Attributions Revision and Adaptation. Example 2 List the multiplicities of the zeroes of each of the following polynomials. To improve this estimate, we could use advanced features of our technology, if available, or simply change our window to zoom in on our graph to produce the graph below. The factor theorem leads to the following fact. This example leads us to several nice facts about polynomials. Only polynomial functions of even degree have a global minimum or maximum. Here is the first and probably the most important. We can see the difference between local and global extrema below. Those require a little more work than this, but they can be done in the same manner.

For now, we will estimate the locations of turning points using technology to generate a graph. 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. Again, if we go back to the previous example we can see that this is verified with the polynomials listed there. Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a polynomial function written in factored form will have an x-intercept where each factor is equal to zero, we can form a function that will pass through a set of x-intercepts by introducing a corresponding set of factors.

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