Find all complex zeros of each polynomial function. Give exact va...

Question

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=5x^3-9x^2+28x+6

Accepted Answer

Welcome back. I am so glad you're here. We're told for the following polynomial function to determine all the complex zeros, right? The answer in exact form our polynomial function is F of X equals three X cubed minus 10, X squared plus 22 X plus 20. Our answer choices are answer choice, A negative two thirds and two plus the squared of six. I answer choice B negative two thirds and two minus the squared of six. I answer choice C two thirds, two plus squared of six I and two minus squared of six. I answer choice D negative two thirds, two plus squared of six I and two minus squared of six. I. All right. We've got a degree of three here. And when we've got a degree of three, we first look at it, see if we could possibly factor by grouping. But in this case, it's not going to work. And so if we can't factor by grouping, the next best thing is to work with the rational zeros theorem and figure out all of the possible rational zeros. And in order to do that, we're going to take the factors of the constant term 20 that's P and divide those by the factors of the leading coefficient. In this case, three, that's Q. And let's figure out what all of those possible factors are. For P for 20 we've got a plus or minus 20 which would be multiplied by a plus or minus one. We've got a plus or minus 10 which would be multiplied by plus or minus two. And we've got a plus or minus five, which would be multiplied by a plus or minus four. So lots of possibilities for P for Q factors of three, that's plus or minus three and plus or minus one. So what we do with all of those different combinations is we're going to test them out with the remainder theorem. And we set up synthetic division with all of the leading or with all of the coefficients in our constant term from our polynomials. So we'll have three negative 10, 22 20. And we'll divide all of those by all the possible combinations of P divided by Q. So we could start with 23rd. However, this is multiple choice and all of the real zeros listed in our multiple choice solutions are either negative two thirds or positive two thirds. So because answer choice A has got negative two thirds, we can just start with that and then we can go ahead and do the synthetic division. See if we get a remainder of zero, we bring down the three. And then we multiply negative two thirds multiplied by three. And that is a negative too. Then we add negative 10 plus negative two. That's negative 12. Multiply negative two thirds multiplied by negative 12. It's positive eight add 22 plus eight is 30. Multiply negative two thirds multiplied by 30 is negative 20 add 20 plus negative 20 is zero. And yes, thank you. Remainder theorem negative two thirds is one of our zeros here. So now we've got a new polynomial. We've got a three X squared minus 12 X plus 30 all equal to zero. And with that, we see that everything here is divisible by three. So we can go ahead and divide everything by three that will give us X squared minus four X plus 10 equals zero. So now we see if we could factor this and we cannot easily factor that. So when it's not easily factor, we can use the quadratic formula. We recall from previous lessons, the quadratic formula of negative B plus or minus the square root of B squared minus four AC all divided by two A. And in this case, our A is equal to one, our B in front of that X is equal to negative four and our C is equal to 10. So filling all of that in, we've got a negative negative four which is a positive four plus or minus the square root of B squared, that's negative four squared. That's a positive 16 minus four, multiplied by A, that's one multiplied by C that's 10 all divided by two, multiplied by A, that's two multiplied by one. So that's two simplifying inside of the radical 16 minus four, multiplied by 10. 4, multiplied by 10 is 40. So what's 16 minus 40? That's a negative 24. So we've got four plus or minus the square root of negative 24 all divided by two. Let's simplify that square root of negative 24. Recalling from previous lessons that when we've got a negative in there, that's going to be an I that comes out. So there's an I coming in front and that's the square root of a positive 24 which is the same as six multiplied by four. And we can take the square root of four. That's two. So two comes out, we've got two, I multiplied by the square of six. So in our numerator four plus or minus two, I square at six, all divided by two. Every term here, we can divide by two. So that's going to leave us with two plus or minus I square root of six. Well, we can put that squared of six first and then just noting that the I is not underneath it. And that's written in our correct complex number format of A plus B I. So we've got three zeros here. We have negative two thirds, two plus square at six I and two minus square at six. I, we look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=5x^3-9x^2+28x+6

Key Concepts

Complex Zeros

Polynomial Functions

Finding Zeros