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+2x^2+1

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 X raised to the fourth power plus four, X squared plus four. Our answer choices are answer choice. A negative square root two, I with a multiplicity of four answer trace B square it two, I with the multiplicity of four answer choice C negative square root two, I with the multiplicity of three and square root two, I with the multiplicity of one and answer choice D negative square root two, I with a multiplicity of two and square root two, I with a multiplicity of two. All right. So when we are finding the zeros for this function, we notice that X rays to the fourth plus four, X squared plus four, that's factor. So we set up two open sets of parentheses and we can do the reverse of the foil process. We ask ourselves what multiplied by what gives us x-rays to the fourth power. Those are our firsts. That would be X squared multiplied by X squared. Then we ask ourselves, what multiplied by what gives us a positive four, those are our lasts. Keeping in mind that when we take our outsides multiplied together and add those to our insides multiplied together. That is needs to add up to a positive four X squared. So for our last, in other words, what two factors of positive four add up to positive four, that's going to be plus two plus two. So now to find our zeros, we need to set our factors equal to zero. And well, they're the same thing. So we just need to set X squared plus two equal to + time and solve. Keeping in mind that whatever we find will double its multiplicity. So for X squared plus two equals zero, we are going to subtract two from both sides, we get X squared equals negative two. We take the square root of both sides and we have on the left X that equals plus or minus the square rut of negative two. We can do a little bit of simplifying with the square rut of negative two. We know that that is the same as the square rut of negative one multiplied by two, which is the same thing as the square re of negative one multiplied by the square rut of two. And we recall from previous lessons that by definition, the square rut of negative one is I. So we have I multiplied by the square root of two and keeping in mind that in standard form, the I comes after the number. So we've got the square of two. I just noting that, that I is not inside of the radical. And also noting that that was a plus or minus the square root of two I. And keeping in mind that that was just for one of the factors, but we have another identical one. So we have a multiplicity of two, both for the square root of two I and for the negative square root two, I, so we look at our answer choices and that 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)=x^4+2x^2+1

Key Concepts

Complex Zeros

Factoring Polynomials

The Fundamental Theorem of Algebra