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)=3x^4+2x^3-8x^2-10x-1

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 the given function F of X is equal to six X raised to the fourth power plus 101 X cubed minus 17 X squared minus three X minus 10. Here we have four answer choice options, answer choices A through D, each of which vary in the possible numbers of positive real zeros, negative real zeros and non-real complex zeros. So before we begin this problem, we first want to note that we will be utilizing Descartes rule of signs to help us determine all of our different possible zeros. So starting with the possible numbers of positive real zeros, we want to look at our given function F of X and we will need to determine the total number of sign variations because it will tell us the maximum number, maximum possible number number of positive real zeros. So starting with our first term, we see this term is positive and then we go to our second term. So we go from a positive to our second term which is also positive. So there is no sign variation. So far, we then go from this positive to the next term which is negative. So this is our first sign variation. Next we go from a negative to a negative. And lastly, we go from this negative to our last term, which is also negative. So there is no additional sign variation. So since we only have a sign variation of one, this tells us that our maximum possible number of positive real zeros is just going to be one. And now we know we cannot subtract two from our value of one here. Since that will result in a negative value and we cannot have a negative number of real zeros here. So we can only ever have one positive real zero. So now we can move on to the possible numbers of negative real zeros. So for this, we repeat a similar process where we now assess the sine variation in F of negative X. So substituting in negative X into our given function, we have six X raised to the fourth power minus 101 X cubed minus 17 X squared plus three X minus 10. So now once again counting our sign variations, we start with our first term which is still positive. We now go to the second term which is now negative. So this is our first sign variation. We then go from this negative to the next term which is also negative. Next we go from a negative to a positive. So this is an additional sign variation. And lastly, we go from a positive to our last term, which is negative. So this is our third sign variation. So now we can conclude that the maximum number of negative real zeros is going to be three because that is our total sine variation for F of negative X. So now with this total of three, we can subtract two which gives us one. So we either have three or one negative real zeros. So now we can move on to the last part of our problem where we need to determine the possible numbers of non-real complex zeros. So for this, we need to look at the degree of our polynomial which we see is four. So again, we have a degree of four. So this tells us that we must have four complex zeros. So we need to go through all of the combinations of positive and negative real zeros to determine our non-real complex zeros. So starting with our first combination here we have one positive real zero and then we will have three negative real zeros. And then we will have some value for our non-real zeros. And then all of these values must sum to the degree which is four. So we have one plus three plus some value is four. And here the missing value is zero. So we can have zero non-real zeros. Now testing our second and final combination, we will have one positive real zero and one negative real zero plus sum value of non-real zeros and they all the sum to four. So in this case, the missing value is just two. So with this, we can conclude here that our possible numbers of non-real complex zeros are either zero or two. So by utilizing de cartes rule of science, we have found all of our possible numbers of zeros. And so we are left here with answer choice C where again, the possible numbers of positive real zeros are only ever one. The negative real zeros are three or one and the non-real complex zeros are zero or two. 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)=3x^4+2x^3-8x^2-10x-1

Key Concepts

Fundamental Theorem of Algebra

Descarte's Rule of Signs

Complex Conjugate Root Theorem