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^5-7x^3+6x+8

Accepted Answer

Hey, everyone. Here we are asked to determine all of the possibilities for the numbers of positive negative and non-real complex zeros from the given function F of X is equal to four X to the fifth power minus seven X cubed plus nine X plus 15. Here we have four answer choice options. Answer choice. 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 non-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 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 by finding the number of possible positive real zeros by assessing the amount of sine variations in our given function F of X. So here looking at F of X, we see our first term is positive and then our next term is negative. So we go from a positive to a negative term. So this is one sine variation and then we go from this negative term to another positive term. So this is our second sign variation. And lastly, we go from the positive term to our last term, which is also positive. So there is no additional sign variation. So in summary, we see our variation, our V variation for our function F of X is two. And so because of this two, this two tells us that we have a possible number of positive real zeros of two. But if we take this value of two and subtract two, we see we end up with zero. So this tells us we either have two or zero, real positive zero. So now we can find number of possible negative real zeros by assessing the sine variation in F of negative X. So substituting in negative X into our function, we see that all of the terms with odd powers have a sign change. So we see our first term four X to the fifth power becomes negative four X to the fifth power. Then our second term we have plus seven X Q, then we now have negative or minus nine X. And lastly, our last term is a constant. So it remains the same as just plus 15. So now we can note these sign variations. So going from our first term, we now go from a negative term to our second term which is positive. So this is one sign variation. Then we go from this second term, which is positive to the third term, which is negative. So this is our second sign variation. And now moving to the next term, we go from a negative to a positive of our last term. And so we see we have an additional third variation. So now summarizing this, we see that the variations in our function F of negative X is a total of three variations. So this tells us that we could have three possible negative real zeros. But if we take this three and subtract two, we get one. So we see that we could either have have three or one negative real zeros. And now for the complex zeros, we just need to note that the degree of our polynomial is five. So this tells us that we must have five complex zeros. So in summary, we see that the number of possible positive real zeros is two or zero. And then the number of possible negative real zeroes is three or one. And then the number of possible non-real complex zeros. Since again, our degree is of five and complex zeros or roots come in pairs, we have the possible numbers of zero or four. So now with our possibilities, summarized, we see that we are left with answer choice D where positive real roots are two or zero, negative real roots are three or one and the non-real complex zeros are zero 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^5-7x^3+6x+8

Key Concepts

Fundamental Theorem of Algebra

Descarte's Rule of Signs

Complex Zeros