Determine the different possibilities for the numbers of posit...

Question

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

Accepted Answer

Hey, everyone. Here we are asked to find all of the possible numbers of positive negative and non-real complex zeros. For the given function F of X is equal to three XX raised to the power of five, five minus 34 X raised to the power of four plus 57 XQ minus 11 X squared plus two X plus nine. 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 begin this problem, we can start by finding our possible numbers of positive real zeros and we can do this by recalling Descartes rule of signs which allows us to utilize these sine variations in F of X to determine our numbers of positive real zeros. So assessing these sign variations in our given function, we start with the first term which is positive, we then move to the second term which we can see is negative. And so we go from a positive to a negative. So this is one sign variation, we then go from our second term to the third term which is a negative to a positive. So this is an additional sign variation, then we go from this positive to a negative. So once again, we have an additional sign variation. Next, we go from this negative to a positive. So this is 1/4 sign variation. And lastly, we go from a positive to our last term, which is also positive. So there is no additional sign variation. So since we have four sign variations, this tells us that our maximum number of possible number of positive real zeros is four. And now we take this value of four and we can subtract two which gives us two. And this tells us we could have four or two positive real zeros. And now from two, we see we can subtract an additional two which gives us zero. So we could have either four or two or zero numbers of positive real zeros. So now we can move on to determining our numbers of negative real zeros. And so we repeat a similar process. However, this time, we need to assess the sign variation of F of negative X. So substituting in negative X into our given function, we get negative three X raised to the fifth power minus 34 X raised to the fourth power minus 57 X cubed minus X squared minus two X plus nine. So now looking at our sine variation for F of negative X, we see we go from a negative to a negative to another negative to again, one another negative and to another negative. And lastly going from the second, last to the last term, we go from a negative to a positive term. So we see we only have one sign variation for F of negative X. So this tells us that our maximum number of negative, our maximum number of negative real zeros is just one. And here we see we cannot subtract two from one since this will result in a negative value. So we can only have one negative real zero as a possibility. So now we can move on to our last part of the problem where we need to find the possible numbers of non-real complex zeros. So the first step is to first note the degree of our polynomial. Here we see that the degree is five. And so this tells us that we must have five complex zeros. So now we can test out our different combinations of real zeros to determine our amount of non-real complex zeros. So for example, we know we could have four positive real zeros. We also can only have one negative real zero. And then we will have some value of non-real complex zeros. And all of these values must some to our degree which is five. So we have four plus one plus some value gives us five. And here this value is zero. So one of our possible numbers of non-real complex zeros is just zero. So after testing all of the combinations, we find that the numbers of non real complex zeros come out to either 02 or four. So now we have found all of the possible zeros. And so we are left here with answer choice. A where again the numbers of positive real zeros are either four or two or zero. The negative real zeros are only ever just one 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. $ƒ (x) = 2 x^{5} - x^{4} + x^{3} - x^{2} + x + 5$

Key Concepts

Fundamental Theorem of Algebra

Descartes' Rule of Signs

Complex Conjugate Root Theorem

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. ƒ(x)=2x5−x4+x3−x2+x+5ƒ(x)=2x^5-x^4+x^3-x^2+x+5

Key Concepts

Fundamental Theorem of Algebra

Descartes' Rule of Signs

Complex Conjugate Root Theorem

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