Determine the different possibilities for the numbers of posit...

Question

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=11x⁵-x³+7x-5

Accepted Answer

Hey, everyone. Here we are asked to find all of the possible numbers of positive negative and non-real complex zeros. For the given polynomial function F of X is equal to nine X raised to the fifth power minus 31 X Q plus five X plus seven. Here we have four answer choice options, answer choices A through D, each of them stating the total numbers of positive real zeros, negative real zeros and non-real complex zeros. So to begin this problem, we can start by finding the numbers of positive real zeros. And to do this, we need to recall Descartes rule of science which states that for positive real numbers, we can just use the given function and count the numbers of sign changes. So here going from the first term to the second term, we go from a positive to a negative. So this is one sign change, we then go from a negative to a positive. So this is a second sign change. And then lastly, we go from a positive to another positive value. So there is no additional sign change. Well, so here we see that we have a total number of two sign changes. So that tells us we have a total number of two possible positive real zeros. However, recalling Descartes rule of signs, we know that we can also subtract two from this total value and then that gives us zero. So this tells us that we can actually have two or zero, positive real zeros. So now we can move on to finding the numbers of negative real zeros. And for this, we repeat a similar process. However, we need to count these signs of F of negative X. So first, we need to find F of negative X by just substituting in negative X into our given function. In doing this, we get a new function here. A new expression negative nine X raised to the fifth power plus 31 X cubed minus five X plus seven. So now once again, we need to count the number of sign changes. So starting with the first term and going to the second term, we go from a negative to a positive. So this is one sign change, we then go from a positive to a negative. So this is, is a second sign change. And then lastly, we go from a negative to a positive. So this is a third sign change. So in total here we see we have three sign changes for F of negative X. And that tells us we have a total of three possible numbers of negative real zeros. However, just like previously, we know we can subtract two from the three here, the total number and that gives us one. So this tells us we have three or one possible numbers of negative real zeros. So now our last step is to find the numbers of non-real complex zeros. So the first step in this is to determine the degree of our polynomial which we see here is five. So again, we have a degree of five. And now recalling the fundamental theorem of algebra, we know that because we have a degree of five, we must have five complex zeros. So we know that the sum of the possible number of positive and negative real zeros as well as the non-real complex zeros will equal five. So using this value of five, along with a combination from earlier of our positive and negative real zeros, we can find our complex zeros, the non-real complex zeros. So we can use one combination where we had two positive real zeros. And then we have plus three negative real zeros. And then we have plus some number of non-real complex zeros. And this sum to uh should equal the degree which is five. So we know that two plus three plus zero gives us five. So that tells us one of the possible numbers of non-real complex zeros is going to just be zero. And now repeating this process testing all of the combinations of positive and negative real zeros. We find that we have multiple possibilities of non-real complex zeros which are zero as we found. And we also have two and we also have four. So now with all of the information that we found, we see that we are left here with answer choice D where again, the positive real zeros are two or zero, the negative real zeros are three or one and the non-real complex zeros are 02 or four. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=11x⁵-x³+7x-5

Key Concepts

Fundamental Theorem of Algebra

Descartes' Rule of Signs

Complex Conjugate Root Theorem

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=11x5-x3+7x-5

Key Concepts

Fundamental Theorem of Algebra

Descartes' Rule of Signs

Complex Conjugate Root Theorem

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=11x⁵-x³+7x-5