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)=x^4-8x^3+29x^2-66x+72

Accepted Answer

Hello, today we're gonna be solving for the zeros for the following polynomial function. Now, since we want to solve for the zeros, we're going to set the polynomial function equal to zero. But the issue here is that we have a higher order polynomial. So how do we simplify a higher order polynomial? Well, in order to do that, we can go ahead and take the PQ approach, which is also known as the rational zero approach. In order to use this approach, we're going to write out all the possible factors for the last term which are going to be the P factors and all the possible factors for the first term which are going to be the Q factors. Now, the factors for the first term are pretty easy to determine the leading coefficient of the first term is one and the only multiple that can give us one is just one. But what about the factors for the last turn? Well, the last term is 24 and all the possible factors for the last term is going to be 123468, 12 and 24. Now, what we're going to do is, we're going to set up our PQ factors. So we're going to put all the P factors over all the Q factors and these factors always come out as positive or negative. So we're going to have 123468, 12 and 24 all over one. But since these numbers are all over one, that means the simplified form of the PQ factors are going to be this long list. So in order to simplify the current polynomial function, we want to go ahead and use synthetic division, setting up our synthetic division, we're first going to draw the synthetic division symbol. Then what goes inside this region is going to be all the coefficients from the polynomial function that is going to be one negative 8, 26 negative 4024. Now, what we want to go ahead and do is we want to pick one of the possible zeros from the PQ factors to put into our synthetic division and see whether that possible zero gives us a remainder of zero. Well, let's go and try, go ahead and try the PQ factor of two. So in, in order to do the synthetic division, we're going to bring down the first term, which is one, then what goes under the next value is going to be the current value times the possible zero that we chose. So what's going to go under negative eight is going to be one times two which is going to give us two, negative eight plus two is going to give us negative six. Then we're going to repeat this process of taking the current value and multiplying it by a possible zero. So negative six times two is going to give us negative 12 and 26 minus 12 is going to give us positive 14, 14 times two is going to give us 28 and negative 40 plus 28 is going to give us negative 12 and finally negative 12 times two is going to give us negative 24 and 24 minus 24 is going to give us zero. So since we have a remainder of zero, that means that the possible zero, which in this case is X is equal to two is a possible zero for simplifying the polynomial function. But we want to represent this possible zero as a factor of X. So we're going to have to set this value equal to zero. In order to do that, we can subtract two to both sides of this quantity that is going to give us X minus two is equal to zero. And that means that the possible zero which is two can be represented by X minus two. Now we're gonna multiply this quantity by the simplified form of our polynomial function, we're gonna write it using the new values as leading coefficients in the simplified polynomial. And the first term of our polynomial is going to be one degree less than the original. So we can rewrite the simplified polynomial as X cubed minus six X squared plus 14, X minus 12. And this is one of those cases where we're going to have to do synthetic division one more time in order to further simplify the polynomial. So let's go ahead and set up the synthetic division once again. So we're gonna list the new polynomial coefficients which is going to be one negative 6, 14 and 12. Then we want to go ahead and pick another possible zero from the PQ factors that will produce a remainder of zero. Now it's OK to pick repeating values of the zeros. Sometimes you will end up with repeating values. So let's go ahead and try positive two. Once again, we're going to bring down the first term which is one and one times two is going to give us positive two, negative six plus two is going to give us negative four and negative four times two is going to give us negative 8, 14 minus eight is going to give us positive six and six times two is going to give us 12. Finally negative 12 plus 12 is going to give us zero, which gives us a remainder of zero. So that means we ended up with another zero of X, which is X is equal to two. And just like before, we can represent this value as X minus two is equal to zero. So putting our new factor together, this is going to give us X minus two times X minus two times our new simplified polynomial, which is going to be X squared minus four X plus six. Now that we've simplified the polynomial function, we can go ahead and use our zero property to set each of the factors of X equal to zero. That is going to give us X minus two is equal to zero, X minus two is equal to zero and X squared minus four, X plus six is equal to zero. Now, for these two values, we know that we're going to get X is equal to two, but we're going to get it two times. What this means is that the value X is equal to two is going to have a multiplicity of two because it's going to pop up twice as a possible zero. Now, for the other factor of X, which is X squared minus four, X plus six, this is not a factor trinomial, but we can solve this using the quadratic equation. So the quadratic equation is going to give us X is equal to negative negative four, which is going to give us positive four plus or minus the square root of negative four squared minus four times one times positive six all over two times one. Now simplifying the inside is going to give us four plus or minus the square root of 16 minus 24 all over two and 16 minus 24 is going to give us the value of negative eight. So currently X is equal to four plus or minus the square root of negative 8/2. Now keep in mind whenever you have a negative number inside the square root, that is going to produce an imaginary value. So that means that X is going to equal to four plus or minus the square root of eight I all over two. Now, the square root of eight is not a perfect square. However, it can't be simplified as two times the square root of two. So we have X is equal to four plus or minus two square root of two I all over two. And notice that in the numerator, we can go ahead and factor out a multiple of two from the leading coefficients. So we get X is equal to two times two plus or minus the square root of two I all over two. And since we have two in both the numerator and denominator, we can just go ahead and cancel out those values leaving us with X is equal to two plus or minus the square root of two. I. Now this is going to give us two more zeros for X. That is going to tell us that X is equal to two plus the square root of two, I and X is equal to two minus the square root of two I. And so just to summarize everything, we have four possible zeros for X or I'm sorry, four complex zeros for X in total, we have X is equal to two with the multiplicity of two. It's equal to two plus the square root of two I and two minus the square root of two I. Since these are the four possible complex zeros for X, that means that the answer to this problem is going to be B So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x^4-8x^3+29x^2-66x+72

Zeros of Polynomial Functions practice set