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

Accepted Answer

Hey, everyone. Here we are asked to determine all of the possibilities of the number of positive negative and non-real complex zeros. For the given function F of X is equal to three X to the fifth power minus 34 X to the fourth power plus 57 X cubed minus 11 X squared plus two, X plus nine. Here we have four answer choice options. Answer a positive real zeros four or two or zero, negative real zeros, one non-real complex zeros zero or two or four. Answer B positive real zeros two or zero, negative real zeros one and not real complex zeros two or four. Answer C positive real zeros, one negative real zeros three or one non-real complex zeros one or three. And 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 this problem by first determining the possible number of positive real zeros. And we do this by first assessing the side variation in our given function F of X. So looking at F of X, we see our first term is positive. So we go from our first term, our, our first term to our second term. So we from a positive to a negative. So here we can see we already have one sign variation. Then moving from the second term to the third term, we go from a negative to a positive. So this is our second sign variation, then we go from a positive to a negative. So we have a third sign variation, then we go from a negative to a positive. So we have 1/4 sine variation and lastly going to our last term, it is just a positive to another positive. So we do not have an additional sine variation. So now just summarizing, we see that the variations in F of X are a total of four. So this value of four tells us that we have a possible number of four positive real zeros. And well, if we take this value of four and subtract two, we get two. So we could also have just two possi possible number of real zeros that are positive. And then if we take this original number of four again and subtract four, we get zero. So in summary, we can have four or two or zero, positive real zeros. So now we can determine the possible number of negative real zeros by finding first the sine variation in F of negative X. So to find F of negative X, we just substitute in negative X into our original function F of X. So we see that all, all of our terms with odd powers will flip their sign and then the terms with even powers will just remain the same. So starting with our first term, we had positive three X to the fifth power. Well, for F of negative X, we now have negative three X to the fifth power. Then our second term stays the same. So we just have minus 34 X to the fourth power. Then for our third term, we now have minus 57 X cubed, then we have minus 11 X squared. Next we have minus two X and lastly, we still just have this plus nine. So now we just need to note these sign variations for F of negative X starting again with our first term. And going to our second term, we go from a negative to a negative. Then moving forward, we go again from a negative to a negative. Then from our third term to our fourth term, it is again a negative term to another negative term. And then from there we go again from a negative term to an additional negative term. But going from our second to last term to our last term, we go from a negative to a positive. So see here we see that we only have one sign variation. So summarizing the variations for F of negative X are just one. Well, this value of one tells us that we simply have one possible number of negative real zeros. So now to determine our complex zeros, we need to note the degree of our polynomial. And here we see that the degree of our polynomial is five. So this tells us that we must have five complex zeros. So now summarizing our findings, we see that the number of positive real zeros, we can have either four or two or zero. And then the number of negative real zeros, we can only have just one. And then lastly for our possible number of non-real complex zeroes, we have either zero or two or four. So with all of our findings summarized, we are left here with answer choice. A where again, we have positive real zeros, four or two or zero, negative real zeros one 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)=2x^5-x^4+x^3-x^2+x+5

Key Concepts

Fundamental Theorem of Algebra

Descarte's Rule of Signs

Complex Conjugate Root Theorem