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

Question

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=5x^4+3x^2+2x-9

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 function F of X is equal to four X raised to the fourth power plus five X squared plus five X minus three. Here we have four answer choice options, answer choices A through D, each of them stating some amount of positive and negative real zeros as well as the possible number of non-real complex zeros. So to begin this problem, we can start by finding our numbers of positive real zeros. And so we first need to recall Descartes rule of signs which states that we can simply count the number of sign changes for our given function F of X to find the amount of positive real zeros. And so here going from the first term to the second term, we see we go from a positive to an another positive. So there is no sign change. Then the same thing happens from the second term to the third term. However, from the third term to the fourth term, we go from a positive to a negative. So we see that there is one sign change. And from this single sign change, this tells us that we have one possible number of positive real zeros. And so now we can move on to finding the possible number of negative real zeros. And so we re we repeat a similar process. However, this time we count these sign changes in F of negative X. And so we just need to substitute in negative X into our original function. So we have four multiplied by negative X raised to the fourth power plus five, multiplied by negative X raised to the power of two plus five multiplied by negative X minus three. Now we just need to simplify our expression here. So we see that F of negative X simplifies 24 X raised to the fourth power plus five X squared minus five X minus three. Now going term by term here from the first term to the second term, there is no sign change since they are both positive, then we go from the second term to the third term. And we see here we go from a positive to a negative. So this is one sign change and then we go from a negative to a negative. So we see in total there is just one sign change or F of negative X. Well, this tells us that we have only one possible number of negative real zeros. And so finally, we need to find our possible number of non-real complex zeros. And So to do this, we first need to note the degree of our polynomial, which we see is four. So again, our degree here is four. And as we recall from the fundamental theorem of algebra, this degree of four tells us that we must have four complex zeros. So this tells us we have a sum of four complex zeros, which means using our information from earlier, we know we have one positive real zero, we have one negative real zero. And then we have some number of non-real complex zeros, but they all need to sum to the degree of four. Well, we see here that this missing number for the non-real zeros is going to be two since one plus one plus two sums to four. So we know that we again have two non real complex zeros. And so with this, we have found all of our possibilities and we are left here with answer choice C where again, we have one possible number of positive real zeros. We have one possible number of negative real zeros and we have two possible numbers of non-real complex zeros. 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)=5x^4+3x^2+2x-9

Key Concepts

Fundamental Theorem of Algebra

Descarte's Rule of Signs

Complex Conjugate Root Theorem