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^3-4x^2+2x+7

Accepted Answer

Hello, everyone. We are asked for the given function by the various potential scenarios for the count of positive negative and non-real complex zeros. The function we are given is F of X equals nine X cubed minus 13 X squared plus seven X plus one. We have four answer choices with varying amounts of positive negative and non-real complex zeroes. To begin with, we should find how many total zeros we expect from this problem. We can do that by looking at the degree of the function we find the degree by looking at the highest exponent. In this case, it is cubed. So that means it's the third degree. So we know that we will have a total of three zeros when we are done. So starting out by looking at positive real zeros, we find our positive real zeros by counting the sign changes in the F of X function. So sign changes in F of X. So our F of X function is what we were given initially. So I'm not going to rewrite it just yet. Our first term is a positive second term is a negative, third term is a positive fourth term is a positive. So we change from positive to negative. So there's one and then negative back to positive. So two, we have two sign changes, which means we could have two positive real zeros. But recall that if a value of zeros is two or greater, we subtract a factor of two to account for any possible non-real zeros. So for our positives, we could have two real zeros, but we could also have zero. We don't honestly know at this point for negatives, we want to find how many sign changes in the F of negative X function. So I'm going to put negative X in for X in this function and see if it changes how many sign changes we have. So we'd have nine multiplied by negative X cubed minus 13 multiplied by negative X squared plus seven multiplied by negative X plus one. So simplifying this a little bit, a negative number cubed will remain negative. So a negative multiplied by nine, we would have negative nine X cubed for negative X squared. When a negative number is squared, it becomes positive. So this will remain negative 13 X negative X multiplied by seven. We will have a negative, I'm sorry, negative seven X and then still plus one. So checking out the signs we have. Now my first term is negative, second term is negative, third term negative, fourth term positive. So this means we have one sign change. So we possibly have one negative real zero. So writing everything I know down or positives, I know I have two or zero for negatives, I have one. So trying to find complex are non-real complex zeros. Recall that the total has to be three. So if we have two positive zeros and one negative zero, there's no space for a complex zero. So that would be zero. However, if we have zero, positive zeros, one negative zero, then we would have to have two complex zeros. So the, this is our scenarios where we have two or zero, positive real zeros, we have one negative real zero and we have zero or two non-real complex zeros and this matches answer choice D have a nice day.

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=2x^3-4x^2+2x+7

Key Concepts

Fundamental Theorem of Algebra

Descarte's Rule of Signs

Complex Conjugate Root Theorem