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)=6x^4+2x^3+9x^2+x+5

Accepted Answer

Hey, everyone. Here we are asked to find out all of the possible numbers of positive negative and non-real complex zeros. For our given function F of X is equal to X to the fourth power plus three X cubed plus 25 X squared plus three X plus one. Here we have four answers. Choice options, answer choice. A positive real zeros four or two or zero, negative real zeros zero, non-real complex zeros zero or two or four. Answer B positive real zeros zero, negative real zeros four or two or zero. Non-real complex zeros zero or two or four. Answer C positive real zeros, one negative real zeros three or one non-real complex zeros zero or two. And answer choice D positive real zeros, two or zero, negative real zeros two and non-real complex zeros zero or two. So here we can begin by determining the possible number of positive real zeros by first identifying the amount of sign variations in our original function F of X. So looking at our given F of X function, we begin with a positive term for the first term and then we go to the second term which is also positive. And then we see it as we keep going down to the next term, we go to another positive and then we go from a, another positive to another positive. And even with our last term, we go from a positive to another positive. So in summary, here, we can see that the total sine variations for F of X are zero, since we have all of our terms positive, we do not have any sign variation. And so this value of zero tells us that we have zero numbers of positive real zeros. So now we can move on to determining the possible number of negative real zeros. And so for this, we need to just first identify the amount of sine variations in F of negative X. So here, since we are substituting in negative X four X from our original function, we can notice that all of our terms with even powers will remain the same, but the terms with odd powers will have a sign change. So starting with our first term, it will remain as 15 X to the fourth power. But then our second term, since it is an odd power, it will become minus three X to the third power. And then for our third term, it remains the same as plus 25 X squared, then we have minus three X and lastly plus one. So here we see for our sign variations, we start with a positive term and then go to a negative term. So we have one sign variation. And now starting with our second term, we go from a negative term to our third term, which is positive. So this is a second sign variation, then we go from this positive to a negative. So we have a third sign variation. And lastly, we go from this negative term to our last term, which is positive. So we have 1/4 sign variation. So now summarizing, we see that our variations or our function F of negative X is a total of four. Well, this number of four tells us we could have four negative real zeros. Well, if we take this value four and subtract two, we get two. So we could also have two negative real zeros. But if we take the original value four and now, so track four, we get zero. So in summary, we could have four or two or zero, negative real zeros. And now to determine our number of non real complex zeros, we need to note the degree of our polynomial. Well, the highest degree of our function is four. So the degree of our polynomial here is four and this tells us that we must have four complex zeros. So now summarizing our findings, we have the possible number of positive real zeros could be just zero. So we will always have just zero, positive real zeros. And now for the possible number of negative real zeros. We found we could have either zero or two or four. And now for the possible number of non-real complex roots or complex zeros, we have either zero or two or four. So now with our findings summarized, we see we are left with answer choice B where the positive real zeros are just zero, the negative real zeros could be four or two or zero and the negative complex or the non-real complex zeros are zero or two 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)=6x^4+2x^3+9x^2+x+5

Key Concepts

Fundamental Theorem of Algebra

Descarte's Rule of Signs

Complex Conjugate Root Theorem