In Exercises 47–48, find an nth-degree polynomial function wit...

Question

In Exercises 47–48, find an nth-degree polynomial function with real coefficients satisfying the given conditions. Verify the real zeros and the given function value. n = 3; 2 and 2 - 3i are zeros; f(1) = -10

Accepted Answer

Hi there. So for this problem is considering the conditions of an unknown polynomial equals three and one and they do one plus four ir zeros and f equals 50. Find this polynomial. So one thing we know right off the bat is our industry, which means this is going to be a polynomial X. The third or with the 33. So let's go out and buy our formula for X the third. You have a X third plus be X squared plus C, X plus D. I'm also gonna multiply this by some factor A. Just so we can account for the constant. It could be multiplied by and so we'll determine that a bit later. But first we're gonna go ahead and write up our zeros. So we have x equals one and X equals negative one plus four. I now with this with imaginary zeroes were supposed to have two answers for imaginary, we're going to have another zero if you remember um X equals negative one minus fry. So remember for your book if you have imaginary zero you should have another imaginary zero with a different sign from the imaginary number. So in case we have also a negative one minus fry. So our next step is to figure out what our polynomial should be at first. So we're gonna multiply all these zeros together. But first we have to write them as functions or expressions. So we do that. We're gonna set all of them equal to zero and solve. So in other words I'm gonna said move this one, I have X minus one equals zero. So that's one of our factors as x minus one. Similarly we'll do that for the imaginary numbers. I add one subject for I on both sides I get X plus one minus four I equals zero And finally the last one Add one Add 4. I I get X plus one plus four I equals zero. So one thing I'm going to also do is put parentheses around my one plus four. I my one minus four I just to keep them separate from the main thing. That's our imaginary number. We're gonna multiply all of our factors together. So I have X - X-us 1 -4. I and X plus one plus four. I okay and we'll have our day in front of that at the end of that. At the end let's go ahead and multiply these together with foil. So remember foil? Our lesson oil is really multiply our first terms, our outer terms, our inner terms and our last terms. So we'll come and leave the X -1 for now and just work on the imaginary imaginary expressions. So I'm gonna write this ready? You first have X right here times this X. That gives us X squared. Now we're outside. We have X times one plus four. I so we add X times one plus four. I our inside we have one minus four. I times X all go and add X times one minus four. I finally we have plus one plus four I times one minus four. I let's go ahead and simplify this. So we should have our X squared comes down. We're adding X times one which is X. X. Times four I. Which is for I. X. They're adding another X. X. Times one. And we're subtracting or I times X. With this negative four I hear. And now we have another foil with our imaginary so just like before 9-1 times one. One times later for I one and later four times and you can add those two there one times 1 is one, one times four I is minus four I. Four I times one is plus four I and finally four I times negative four. I is not even 16 times I squared. It's gonna simplify this even further. So now we have X square. Now we have an X. Here and the X. Here I think it's just a two X. We have a negative four I. X. And a positive four X. Those are going to cancel each other out. Also over here we have a negative four I and a positive four. I. Those also cancel each other out. Going down are one And we bring down our Native 16 square. Now one thing to also remember is our imaginary I squared is equal to negative one. You also remember that I squared equals negative one. So we have to bring down negative 16 and you can substitute a negative one for the I squared. Finally we can say that this is X squared plus two X plus one plus 16 which goes to X squared plus two X plus 17. So that is the simplification for multiplication. Now we have to multiply again by X -1. So if I bring this back down, I have X -1. And what we just got X squared plus two X plus 17. So we'll go ahead and simplify this now. So we had the foil yet again and this time we have three terms. So we multiply that all three that term to that term to this term and this term to this term to this one right there, it's our first one will be X times X squared which is X cubed. Our next one is X times two X is two X squared x times 17 17 X. Now we have negative one times X squared is negative square negative one times two X negative two X. And finally negative one times 17 is negative 17. That would be simplified even further. We get executed. Um two X squared minus X squared is a positive X squared 17. X -2, X is a 15 x. and finally We just dropped down R -17. Okay, we're almost there. So remember from earlier we had an A in front of this, we wanna check our multiplier for a So I'm gonna put an a in front of this let's say a multiplied by x cubed plus X squared plus 15 X minus 17. And so we're gonna go ahead and use that final piece of information. The problem gave us top, We notice we have f of two equals 50. So we're gonna use that substitution. So We have f of two peoples 50. That means that if we put an X. of two we should get 50 out as our output. So Mexico's too And we'll say f of x equals 50. Let's go ahead and substitute everything we have here. If you know that are FFXR output is 50. You will say 50 equals power. A well supplied by two cubes plus two squared plus times two minus 17. Now I can simplify this. 50 equals are big. A. Well supplied by two cubed is 82 squared is 4, 18 is 30 times two is 30 to 13. 30. and finally 17 -17, add all those together. Eight plus 4, 12 plus 20 is 32, -17. On site it's 42 -17. That should be 25. So we want to simplify this 25 there I should have 50 equals 25. Finally go ahead and divide both sides by 25. Our A is going to equal to. So finally our actual polynomial Will be our two multiplied by X cubed plus X squared plus X minus 17. So this problem simplifies down to to execute Plus two x squared Plus 30 X -30. For once, we simplify our multiplication at the end. Many of that go back up and look. Our answer is answer D. Alright, I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 47–48, find an nth-degree polynomial function with real coefficients satisfying the given conditions. Verify the real zeros and the given function value. n = 3; 2 and 2 - 3i are zeros; f(1) = -10

Key Concepts

Polynomial Zeros and Their Multiplicity

Complex Conjugate Root Theorem

Using Given Function Values to Find Leading Coefficient