Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x^4+2x^3-3x^2+24x-180

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Welcome back. I am so glad you're here. We're asked for the following polynomial function to determine all the complex zeros. And to write the answer in exact form, our polynomial function is F of X equals X to the fourth plus three, X cubed minus 20 X squared plus 24 X minus 224. Our answer choices are answer choice. A negative 74, negative two square it, two I and two square it two, I answer choice B seven negative four, negative two square, it two I and two square root two, I answer choice C negative and two square root two I and answer choice D negative 74 and negative two square it two I, all right. We have a polynomial function with a degree of four. And our best bet when we've got 1/4 degree is to start with a rational zeros theorem. The rational zeros theorem is going to find all of the possible rational zeros by taking P which is the factors of our constant term negative 224 and dividing that by Q which is all of the factors of our leading coefficient in this case one, well, the factors of the leading coefficient Q one, that's easy. That's plus or minus one. That's great. The factors of our constant term are a bit more numerous. So for negative 224 we've got plus or minus 224 which would be multiplied by plus or minus one. We've got, we know that's divisible by two. So plus or minus two and that would give us multiplied by plus or minus 112. We know that if we took 112 and divided that by two, that would be plus or minus 56. So that's multiplying 56 by plus or minus four to get back to 224. Well, breaking down 56 we know that that is divisible by seven. So this is plus or minus seven. If we take 224 and divide it by seven, we get a plus or minus for and we know 32 56 those are divisible by eight. So we've got a plus or minus eight factor in here. 224 divided by eight will give us a plus or minus 28. And what have we got left? Well, this is also divisible by 14. So we have plus or minus 14 and that would be 14 times 16. So we have plus or minus 16. Those are all the factors of 224 well, negative 224. And because our Q is plus or minus one, any of those divided by one or negative one will get themselves or the negative of it. So we don't need to worry about that part. Now, what do we do with all of this? We take our possible rational zeros and we use the remainder theorem to utilize synthetic division. And we'll see if we get a remainder of zero. And that will tell us we have a real zero. So if we take our coefficients in constant terms from our polynomial, that'll be 13, negative 2024 negative 224. We use synthetic division and we can start, we have so many possibilities here, we could take 224 divided by one which is 224. However, this is multiple choice and the possibilities here are negative seven, positive 74 and negative four. Those are the only ones that are listed for our multiple choice. So let's start there. Answer choice. A has a negative seven. So let's try negative seven. All right. Synthetic division, we bring down the one and then we multiply negative seven multiplied by one, gives us negative seven. And then we add three plus negative seven is negative four, multiply negative seven multiplied by negative four is positive 28 add negative 20 plus 28 is positive eight, multiply negative seven multiplied by eight is po or excuse me, negative 56 add 24 plus negative 56 is negative 32 multiply negative seven multiplied by negative 32 is positive 224. Add negative 224 plus 224 is zero. Yay, thank you. Remainder theorem. And thank you. Multiple choice. We found one of our zeros. It's negative seven. And so now we've got this new polynomial with coefficients in a constant term of one negative 48, negative 32. And so this narrows down our P possibilities because it's only the multiples of 32. So we can eliminate the 224. We can eliminate the 112 the 56 the plus or minus the plus or minus the seven, the plus or minus of 28 the plus or minus of the, let's see the 14. And so now we have fewer possibilities and noting from our multiple choice any time there's a negative seven is paired with a positive four. So might as well be worth it to try 44 synthetic division. So let's do that. Bringing down the 14 multiplied by one is a positive four, add negative four plus four is zero, multiply four, multiplied by zero is zero, add eight plus zero is eight multiply four multiplied by eight is 32. Add negative 32 plus 32 is zero. Yes, we have another zero. It is positive four. Great Well, now what are we left with, we now have coefficients in a constant term of a one X squared. So that's X squared plus zero X is, we don't have to write that part then plus the constant term eight, all equal to zero. Well, we can solve for X like this, we can subtract eight from both sides of the equation. Leaving us with X squared equals negative eight, we can take the square root of both sides. And on the left we have our X that's equal to now plus or minus the square root of negative eight, we know that the square rut of negative eight is equal to the square rut of negative one multiplied by four, multiplied by two. The square root of negative one is I, the square rot of four is two. So we can pull a two I out of the radical, but that's still multiplied by the square of that two that could not escape the radical. So we have plus or minus two I square root two. But we want this in standard form for a complex number so that I will go at the end. So we'll have plus or minus two square root two. I noting that, that I is not underneath the radical. So looking at our answer choices, this matches with answer choice. A 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)=x^4+2x^3-3x^2+24x-180

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Key Concepts

Complex Zeros

Polynomial Functions

Factoring and the Fundamental Theorem of Algebra