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)=-8x^4+3x^3-6x^2+5x-7

Accepted Answer

Hey, everyone. Here we are asked to find all of the possible numbers of positive negative and non-real complex zeros. For our given function F of X is equal to negative 15, X to the fourth power plus X Q minus 12 X word plus five, X minus 16. Here we have four answer choice options. Answer 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 two or one negative real zeros, one non-real complex zeros one or two answer C positive real zeros, one negative real zeros three or one non-real complex zeros zero or two. Answer D positive real zeros two or zero, negative real zeros two and non-real complex zeros zero or two. So here we can begin this problem by first determining the possible numbers of positive real zeros by first assessing the amount of sine variations in our original function F of X. So looking at our given function for F of X, we start here our first term which is negative, then we go to our second term which is positive. So we already have one sign variation, then going from our second term to our third term, we go from a positive to a negative. So this is a second sign variation, then we go from a negative to a positive. So this is our third sign variation. And lastly, we go from a positive to a negative. So we have 1/4 sign variation. So now summarizing, we see that our total variations in our function F of X are four. And so this tells us that we could have four positive real zeros. Well, if we take this value of four and subtract two, we get two. So we could also have two positive real zeros or if we take four and subtract four, we get zero. So we could either have 42 or zero positive real zeros. So now we can move on to determining the possible number of negative real zeros by assessing first the sine variation of F of negative X. So substituting in negative X for X from our original function we see here that all terms with an odd power will have a sign change, but the terms with even power will remain the same. So starting with our first term, it will remain as negative 15 X to the fourth power. Then from our second term, we now have minus X Q, then we have minus 12 X squared, then minus five X and lastly minus 16. So now noting these sine variations starting again with the first term, we go from a negative to the second term which is negative. And we see here as we continue down the line of terms, we keep going from a negative term to another negative term. So in summary, we see that the total number of variations for F of negative X is zero. So this tells us that we will only have zero, negative real zeros. And so now to determine the non-real complex zeros, we need to look at our degree of our polynomial. So the degree of the polynomial is the highest degree of our function, which in this case is four and this degree of four tells us we must have four complex zeros. So summarizing our findings starting first with the possible number of positive real zeros, we found that we could have either 42 or zero, positive real zeros. And now for the possible number of negative real zeros, we saw that this will always be just zero. And now with our degree of four, we can conclude that our possible number of non real of our non real complex zeros, we have either zero or two or four. So now with our findings summarized, we see we are left with answer choice. A where we have the positive real zeros of four or two or zero, negative real zeros of zero and non-real complex zeros 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)=-8x^4+3x^3-6x^2+5x-7

Key Concepts

Fundamental Theorem of Algebra

Descarte's Rule of Signs

Complex Conjugate Root Theorem