Find a polynomial function f of least degree having the graph ...

Question

Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)

Accepted Answer

Or the graph given below, find the polynomial function F of minimum degree. Now let's see what we do in the graph that has two zeros. And we have a bunch of possible answers all with degree of four. So you need a graph with degree of four. We notice here we have two zeros, X equals two and X equals negative two. We can use these zeros to formulate are factors of the polynomial. So we want F of X equals to A for some constant. We don't know multi by X plus two X minus two. And one more thing we noticed is our graph approaches our zeros and bounces off back in the opposite direction. This implies we have double roots or in other words, we have the squared on both of our possible factors. Now let's go ahead and solve for a constant A were given an initial value X is zero, Y is equal to 16. Let's substitute this value into the equation to determine what our A is 16 equals a multiplied by zero plus two squared zero minus two squared. We get 16 equals a multiplied by two squared. Now two squared which gives the 16 equals a multiplied by four and four. Give me a 16 equals a multiplied by 16. And finally, this gives us a equals one. No, we have a function F of X equals X plus two squared, X minus two square. We can solve this by expanding and simplifying our factors. We will get X squared plus four, X plus four for first expansion multiplied by X word minus four, X plus four. Now he wants a foil multi take X squared multiplied by X squared to get X to the fourth X squared and negative four X to get negative four, X to the third X squared and four to get four X squared. Then four X multiplied by X squared to get for the third four X multiplied by negative four X to get negative 16 X squared. Four X multiplied by four to get 16 X. Then four multiply the X squared to get four X squared, four and four X negative to get negative 16 X and finally, four multiplied by four to get 16. We can now combine all of our terms eggs of the fourth stage, the forks, the third plus forks, the third goes away. 16 X in native 16 X goes away. Can we notice we have four X squared minus 16 X squared plus four X squared that gives us negative square and finally bring it down our 16. We just add 16 at the end. You can't simplify any further. So our answer X to the fourth minus eight, X, to the second plus 16 equals to answer D OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)

Key Concepts

Polynomial Function and Degree

Zeros and Their Multiplicities

Local Maximum and Critical Points

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