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^3+6x^2+x+7

Accepted Answer

Hey, everyone. Here we are asked to find all of the possible numbers of positive negative and non-real complex zeros where our given function F of X is equal to 14 X cubed plus 22 X squared plus three, X plus 53. Here we have four answer choice options. Answer a positive real zeros three or one, negative real zeros zero non-real complex zeros zero or two. Answer B positive real zeros two or one negative real zeros zero non-real complex zeros one or two. Answer C positive real zeros zero, 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 one and non-real complex zeros zero or two. So here we can begin by determining the possible number of positive real zeros by first assessing the amount of sign variations in our given function. So looking at our given function F of X, we see that all of our terms are positive. So moving from one term to the next, we see we move from a positive term to another positive term. So summarizing, we see that the total sine variations in our function F of X is zero. Well, this tells us that F of X does not have a positive real zero. So now moving on to determining the amount of negative real zeros in our function, we need to assess the sine variations in F of negative X. So first finding F of negative X, we just substitute in negative X into our given function. And so we will see that all of our terms with odd powers will experience a sign change. However, the terms with even powers will remain the same. So writing our function F of negative X, we have negative 14 X cubed plus 22 X squared minus three X plus 53. So now assessing these sine variations, we see our first term is negative and our second term is positive. So since we are going from a negative to a positive, this is our first sign variation, then we go from a positive to a negative term. So this is a second sign variation. And lastly, we go from this negative term to a positive term, which is our third sign variation. So summarizing, we see that the total sign variations in our function F of negative X is three. Well, this tells us we can have three negative real zeros in our function. However, if we take this three and subtract two, we get one. So this one tells us we can have either three or one negative real zeros in our function. So now to determine the amount of non-real complex zeros, we need to notice the degree of our polynomial here, the highest degree or the degree of our polynomial is three. So this tells us that we must have three complex zeros. So now summarizing everything we have found, we have found that the, the number of possible real positive real zeros, we have found to just be zero. And then the possible number of negative real zeros found to be either three or one. The possible number of non-real complex roots or non-real zeros complex zeros, we have found to be either zero or two. So now with our summary of our findings, we are left with answer choice C where again, we have positive real zeros of just zero, negative real zeros three or one and non-real complex zeros 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^3+6x^2+x+7

Key Concepts

Fundamental Theorem of Algebra

Descarte's Rule of Signs

Complex Conjugate Root Theorem