Find an nth-degree polynomial function with real coefficients ...

Question

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3; 1 and 5i are zeros; f(-1) = -104

Accepted Answer

Hey everyone here we are asked to identify the end degree polynomial function with real coefficients that satisfies the given conditions. So we are first told that n is equal to three. So we know that we have a third degree polynomial function. And with this we need to recall the linear Factory ization theorem for a third degree polynomial which is F of X is equal to a sub N times the quantity of x minus c, sub one times the quantity of x minus C, sub two times the quantity of x minus C. Sub three. And now with this we next just need to identify each of our zeros. So we are told that we have a zero of two. So we can let C sub one, B equal to two. And we also have a zero of four I. So we can let C sub to be equal to this zero of four I. And now since we know that four I is a zero and that our polynomial function has real coefficients. We know that its conjugate negative four, I is also a zero. So we can set it equal to c sub three. And now from here we just need to implement each of these zeros into our formula. So we have F of X is equal to a sub n, times the quantity of x minus C. Sub one, which is two times the quantity of x minus c sub two, which is four, I times the quantity of X minus c sub three, which is negative for I. And now we can just some quick simplifying. So we have F of X is equal to a sub N. Times the quantity of x minus two times the quantity of x minus four I. And finally times the quantity of x minus negative four I. Which gives us X plus four I. And now we want to multiply these last two sets of parentheses together to continue to simplify. So we have again F of X is equal to a sub N times the quantity of x minus two. And instead of boiling out these last two sets of parentheses, we note that these are complex complex contra kits. So so if we were to foil them out, their middle terms would cancel. And also recalling that I squared is equal to negative one. We can actually just square both terms and add them together. So really we have X squared which is squaring the first term and then we have four squared which gives us plus 16. And now with this we can just foil out these last two sets of parentheses. To finish simplifying. So doing so we have F of X is equal to a sub n. Times the quantity of X times X squared which gives us x Q. Then X times 16 gives us plus 16 X then negative two times X squared gives us minus two X squared And -2 times 16 gives us -32. And now we want to start solving for a sub N. And to do this, we need to incorporate our last piece of given information which tells us that F of one is equal to a negative 34. And now with this we can subset to end this one value for X. And then set the function we have equal to this negative 34. So doing this, we have a sub n times the quantity of one cubed plus 16 times one minus two times one squared minus 32 is all equal to negative 34. And now from here, just simplifying by adding these terms, we have negative 34 is equal to a sub end times negative 17. And now getting a sub and alone we divide both sides by -17 and simplifying this. We see that a sub N is equal to positive two. And now we can finish writing our function by substituting this value into our simplified function from before. So we have F of X is equal to a sub N which is now two times the quantity of X cubed plus 16 X minus two, X squared minus 32. And now just factoring in this two to each term, we have a final answer of F of X is equal to two. X cubed plus 32 X minus four, X squared plus sorry minus 64. And now here we can check the accuracy of our solution by graphing this function on a graphing utility and here we can see that we do have a real zero of positive two, and we can also see that at X is equal to one. We do have an output of -34. Therefore, we have an answer up here, which is choice B. We have a function of two X cubed minus four X squared plus 32 x minus 64. Thanks for watching. I hope you found this video helpful, and I'll see you again next time.

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3; 1 and 5i are zeros; f(-1) = -104

Key Concepts

Complex Conjugate Root Theorem

Constructing Polynomials from Zeros

Using Function Values to Find Leading Coefficient