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 determine all of the possible numbers of positive negative and non-real complex zeros. For the given function F of X is equal to 15 X raised to the fourth power plus three XQ plus 25 X squared plus three X plus one. Here we have four answer choice options, answer choices A through D, each of which vary in their possible numbers of positive real zeros, negative real zeros and non-real complex zeros. So to find each of these different zeros, we need to utilize the carts rule of signs. So here we can start with finding our numbers of positive real zeros. So for this, we need to first count the sign variation in F of X to help us determine our possible numbers. So starting with the first term, we see we have a positive term, we then move to, to our second term which is also positive and moving from term to term, we see we keep going from one positive term to an additional positive term. So there are no sign variations in our given function. Well, since there are zero sign variations, this tells us that our numbers of positive real zeros are only ever just going to be zero because we cannot subtract two from this value since this gives us negative two and we cannot have a negative value of zeros. So we just have zero, positive real zeros. So now we can move on to finding our numbers of negative real zeros. So for this, we have a similar process except this time, we need to find these sine variations in F of negative X. So substituting in negative X into our given function, we have 15 X raised to the fourth power minus three X cubed plus 25 X squared minus three X plus one. So now starting back at our first term, we see it is still positive, we then go to our second term which is now negative. So this is our first sine variation in F of negative X. And now we go from this negative to a positive. So this is an additional sign variation. Next we go from the positive to a negative. So this is a third sign variation. And lastly going to our last term, we go from a negative to the last term which is positive. So we have an additional sign variation. So in summary here, the numbers of negative, the numbers of negative real zeroes here, our maximum number is our total sign variation which is four. And now we can subtract two from four, which gives us two. So we can have either four or two possible numbers of negative real zeros. But we can continue to subtract two. So two minus two gives us zero. So now we know we can have either four or two or zero numbers of negative real zeros. So now we can move on to the last part of our problem where we need to find the non-real complex zeros. So for this part, we first need to determine the degree of our polynomial. Here, we see that the degree is four. So since the degree is four, we know we must have four complex zeros. So now we can just test our different combinations of positive and negative real zeros to find the amount of non-real zeroes. So we know we could, we can only have zero positive real zeros. But then we could have, let's say two negative real zeros. And then we will have some value of non-real zeros. And this all the, all these values must sum to the degree which is four. So we have zero plus two plus some value us four. So here we can see the value missing is two and this tells us we could have two non-real complex zeros. So continuing to test each combination, we find that the numbers of non-real complex zeros are either zero or two or four. So with this, we have found all of our different possible zeros. And so we are left here with answer choice B where again, the possible numbers of positive real zeros are only ever zero. The numbers of negative real zeros are four or two or zero and 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