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 identify all of the possible numbers of positive negative and non-real complex zeros. For our given function F of X is equal to six X to the fourth power plus 101 X Q minus 17 X squared minus three, X minus 10. Here we have four answer choice options. Answer choice. A positive real zeros three or one negative real zeros zero non-real complex zeros one or three 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. And 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 by identifying the number of positive real zeros by first assessing the amount of sign variations we have with our given function F of X. So looking at our given function we see with our first term, we have a positive term. And then moving to the second term we see we go from one positive term to another positive term And since there is no sign change, we do not include this as a sign variation. And then from the second term to the third term, we go from a positive to a negative, there is a sign change. So we can identify this as, as our first sign variation. And then we go from a negative term to another negative term and lastly a negative term to the last term which is also negative. So summarizing, we see that the variations in our function F of X, the sign variations are a total of just one. So this tells us that we have one positive real zero in our function. So now to identify the possible number of negative real zeros, we need to find the sine variations in the function F of negative X. So here we just need to substitute in negative X into our original function. And that means that the terms with even powers will have a sign that is unchanged. However, the terms that have odd powers will experience a sign change. So starting with our first term, it will remain as positive six X to the fourth power. And then our second term becomes minus 101 X Q, then we have minus 17 X squared, then we have plus three X and lastly, we have minus 10. So now we have written out our function F of negative X. So we can now account for the sign changes. So going from the first term to the second term, we go from a positive term to a negative term. So this is our first sign variation, then we go from a negative term to another negative term. And then we go from a negative term to a positive term. So this is our second sign variation. And lastly, we go from a positive term to a negative term. So this is our third sign variation. And so now just summarizing, we see that the total variations, the total sign variations in our function F of negative X is three. Well, if we take this full number three and subtract two, we get one. So here we see that we can either have three or one possible numbers of negative real zeros. So now to find our number of non-real complex zeros, we need to note the degree of our function or the degree of our polynomial. So here we see that the highest degree is four. Therefore our degree of the polynomial is four. And this tells us that we must have four complex zeros. So now just summarizing our findings, we see that the possible number of positive real zeros, we had one, we, so we will have one positive real zero. And then the possible number of negative zeros we found can be either three or one. And then we see, since we have a degree of four that the possible number of non-real complex roots or non-real complex zeros, we have the possibilities of zero or two. So now with the summary of our findings, we see here that we are left with answer choice C where again the positive real zeros are 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