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)=x^5+3x^4-x^3+2x+3

Accepted Answer

Hey, everyone. Here we are asked to determine 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 fifth power plus 11 X to the fourth power minus six, X cubed plus 19 X plus plus seven. Here we have four answer choice options. Answer choice. A positive real zeros, two or zero, negative real zeros, one non-real complex zeros zero or four answer B positive real zeros two or zero, negative real zeros, one non-real complex zero, two or four answer C positive real zeros, one negative real zeros three or one non-real complex zeros one or three answer D positive real zeros two or zero, negative real zeros three or one and non-real complex zeros zero or four. So here we can begin by finding our possible number of positive real zeros by first assessing the sign variation of F of X. So looking at our given function for F of X, we have our first term as a positive term. Then moving to our second term, we go from a positive to another positive term. So we do not have a sign variation so far. But going from our second term to our third term, we go from a positive to a negative. So here we have one sign variation, then we go from a negative term to a positive term. So we have a second sign variation and then going to our last term, we go from a positive to another positive. So we do not have an additional sign variation but just summarizing here, we see that our variations for F of X in our case is just two. Well, this value of two tells us that we have a possible number of two real zeros. And so if we take this value of two and subtract two, we get zero. So we can see that we either have two or we have zero positive real zeros. So now we can move on to determining the possible number of negative real zeros by assessing the sign variations in F of negative X. So first finding F of negative X, we see that by substituting in negative X into our original function that all of our terms with odd powers will have their sign flipped, but the terms with even powers will remain the same. So starting with our first term which was originally positive X to the fifth power, it now becomes negative X to the fifth power. Then our second term stays the same. So we have plus 11 X to the fourth power. Then from our third term we now have plus six X Q. Then for our fourth term, we now have minus 19 X and then our last term remains the same as plus seven. So now we want to find our sine variations for F of negative X. So starting again with our first term, we go from a negative to the second term which is a positive. So we see already we have one sine variation, then we go from this positive term to our third term, which is also positive. So there is not a sign variation here. And then we go from our third term to the fourth term. So we go from a positive to a negative. So here we see we have our second sign variation and then lastly, we go from a negative term to a positive term. So we have a third sign variation. So now summarizing, we see that our variations for F of negative X are, are a total of three. So this tells us that we could have three posit possible negative real zeros. And so now if we take this three and subtract two, we get one. So this tells us that we could either have three or we could have one negative real zeros. So now to determine our complex zeros, we need to note the degree of our polynomial. So looking at our function, we see that our highest degree is five. So the degree of our polynomial is five and this tells us that we must have five complex zeros. So now just summarizing our findings, starting with the number or the possible number of positive real zeros, we found we could have either two or zero. And then for our possible number of negative real zeros, we could have either three or one. And now with our degree of five, we can see that we, the possible number of non-real complex zeros can be either zero or four. So now with our findings summarized, we see we are left here with answer choice D where again we have the positive real zeros of two or zero, negative real zeros of three or one and non-real complex zeros of zero or four. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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