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

Accepted Answer

Hey, everyone. Here we are asked to find the possible numbers of positive negative and non-real complex zeros. For our given function F of X is equal to X raise to the fifth power plus 11 X raised to the fourth power minus six XQ plus 19 X plus seven. Here we have four answer choice options, answer choices A through D, each of which vary in the 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 numbers of positive real zeros. And so we do this by recalling Descartes rule of signs which states that we can utilize the sign variation in F of positive X to find our possible numbers of positive real zeros. And so noting the sine variations in our given function F of X, we start with our first term which is positive, we then go to our second term which is also positive. So there is no sign variation so far. But then we go from a positive term to a negative term. So this is our first sign variation. Then we go from a negative to a positive So this is an additional sine variation. And lastly, we go from a positive to a positive. So there is no further signed variation. And so with our total number of variations coming out to two, we can conclude here that our possible number of positive real zeroes, our maximum value here would be two. However, if we subtract two from this value, we get zero. So this tells us that we could either have two or zero possible numbers of positive real zeros. So now moving on to our possible numbers of negative real zeros, we can repeat a similar process. However, this time, we need to note the sine variation in F of negative X. So first substituting in negative X into our given function, we have negative X raised to the fifth power plus 11 X raised to the fourth power plus six XQ minus 19 X plus seven. So now noting the sign variations, we again start with our first term which is negative, then go to our second term which is positive. So this is our first sign variation. We then go from a positive to a positive next, a positive to a negative. So this is an additional sign change. And lastly, we go from a negative to a positive. So this is our third sign variation. So now we can conclude that our maximum number of negative real zeroes is going to be three according to our sign variation in F of negative X. And once again, we can subtract two from this value which gives us one. So this tells us we can have three or one possible numbers of negative real zeros. So lastly, we need to find our numbers of non-real complex zeros. So to do this, we first need to note the degree of our polynomial. And so here the degree is five and this tells us that we must have five complex zeros. And so, since we have five complex zeros, we just need to test our different combinations of positive and negative real zeros to determine our non-real zeros. So for example, we know we can have two positive real zeros, we can also have three negative real zeros. And then we also will have some number of non-real complex zeros. And all of these values must sum to the degree which is five. So we have two plus three plus some value gives us five. Here we see this value is going to be zero. So this tells us that one possible number of non-real complex zeros is going to be zero, trying one more combination here, we could have zero real zeros, positive real zeros, we could also have one negative real zero. And then we will have some number of non-real complex zeros. And this must sum to five here. This value is going to be four for our numbers of non-real complex zeros. And so after testing the combinations, we will find that the possible numbers of non-real complex zeros are 02 or four. So with this, we have all of our possible zeros for our given function. And so we are left with answer choice D where again our possible numbers of positive real zeros are two or zero. Our negative real zeros are three or one and the non-real complex zeros will be 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)=x^5+3x^4-x^3+2x+3

Key Concepts

Fundamental Theorem of Algebra

Descarte's Rule of Signs

Complex Conjugate Root Theorem