# Write a polynomial equation in standard form with the given zeros This is a great check of our synthetic division. Each turning point represents a local minimum or maximum.

### How to write a polynomial in standard form with given roots

We can estimate the maximum value to be around cubic cm, which occurs when the squares are about 2. There is one more fact that we need to get out of the way. We will also use these in a later example. Because a height of 0 cm is not reasonable, we consider only the zeros 10 and 7. The zero factor property can be extended out to as many terms as we need. Those require a little more work than this, but they can be done in the same manner. Even then, finding where extrema occur can still be algebraically challenging. A global maximum or global minimum is the output at the highest or lowest point of the function. Note as well that some of the zeroes may be complex. This fact is easy enough to verify directly. 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. We will use the y-intercept 0, —2 , to solve for a.

We solved each of these by first factoring the polynomial and then using the zero factor property on the factored form. 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. The y-intercept is located at 0, 2.

Do not worry about factoring anything like this. Licenses and Attributions Revision and Adaptation. Find the polynomial of least degree containing all of the factors found in the previous step.

This fact is easy enough to verify directly. The factor theorem leads to the following fact. Sometimes, a turning point is the highest or lowest point on the entire graph.

## How to find the zeros of a function

Also, recall that when we first looked at these we called a root like this a double root. Because a height of 0 cm is not reasonable, we consider only the zeros 10 and 7. Find the polynomial of least degree containing all of the factors found in the previous step. 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. Note as well that some of the zeroes may be complex. These are also referred to as the absolute maximum and absolute minimum values of the function. Each turning point represents a local minimum or maximum. Due to the nature of the mathematics on this site it is best views in landscape mode. It is completely possible that complex zeroes will show up in the list of zeroes. We can estimate the maximum value to be around cubic cm, which occurs when the squares are about 2. Licenses and Attributions Revision and Adaptation. We will use the y-intercept 0, —2 , to solve for a.

Zeroes with a multiplicity of 1 are often called simple zeroes. Here is the first and probably the most important.

## How to write a polynomial in standard form with given roots

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. 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. A global maximum or global minimum is the output at the highest or lowest point of the function. Example 1 Find the zeroes of each of the following polynomials. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Again, if we go back to the previous example we can see that this is verified with the polynomials listed there. Show Solution We will start this problem by drawing a picture like the one below, labeling the width of the cut-out squares with a variable, w.
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