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^4-6x^3+7x^2

Accepted Answer

Hello. Today we're gonna be determining all of the complex zeros for the polynomial function. So what we are given is X to the fourth minus eight X cubed plus 14 X squared. So in order to find the complex zeros, we're going to first set the equation equal to zero. Then what we want to go ahead and do is simplify the polynomial function. Now notice that each of the terms in the function has at least two multiples of X. So we can go ahead and factor out an X squared from the polynomial function that is going to allow us to rewrite the function as X squared times the quantity X squared minus eight X plus 14. Now X squared can be rewritten as X times X. So we have X times X times X squared minus eight X plus 14, equal to zero. Now, what we can go ahead and do is we can use our zero property to set each of the factors of X equal to zero. That is going to give us X is equal to zero, X is equal to zero and X squared minus eight X plus 14 is equal to zero. Notice that the value X is equal to zero pops up two times as a factor of X. That means that the value X is equal to zero is going to have a multiplicity of two since that value of X pops up more than once in the solutions. Now, what we need to do is we need to find the possible zeros of X for the other factor of X. Now X squared minus eight, X plus 14 is not in the form of a factor trinomial. But we can solve for the zeros of X by using the quadratic equation that is going to give us X is equal to negative negative eight which is positive eight plus or minus the square root of negative eight squared minus four times one times 14 all over two times one. Now negative eight squared is going to give us 64 and negative four times one times 14 is going to give us 56. So we have X is equal to eight plus or minus the square root of 64 minus 56 all over two times one which is two. Now 64 minus 56 is going to give us eight. So X is equal to eight plus or minus the square root of eight all over two. Now the eight is not a perfect square, but the square root of eight can be simplified to two times the square root of two. So we have X is equal to eight plus or minus two times the square root of two all over two. Now, in the numerator, we can go ahead and factor out a two. Since both of the leading coefficients are multiples of two, that is going to give us two times the quantity four plus or minus the square root of two all over two. And since we have a multiple of two, in both the numerator and denominator, we can just go ahead and cancel out those quantities that is going to leave us with X is equal to four plus or minus the square root of two. And this is going to give us two individual zeros for X. That means that X is going to equal to four plus the square root of two and X is equal to four minus the square root of two. And this is going to give us two more complex zeros for X. So just to simplify, we have X is equal to zero with the multiplicity of two and we have X is equal to four plus the square root of two and four minus the square root of two. With that being said, 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^4-6x^3+7x^2

Zeros of Polynomial Functions practice set