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Let $y=f(x)$ be a quartic polynomial with leading coefficient $a=1$ and $f(i)=f(2 i)=0 .$ Write an equation for $f .$

$f(x)=x^{4}+5 x^{2}+4$

Algebra

Chapter 2

Polynomial and Rational Functions

Section 5

Zeros of Polynomial Functions

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

Missouri State University

Baylor University

University of Michigan - Ann Arbor

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So we're told that F of I. Is equal to F two I, which is equal to zero. And we're also told that this is a core tech polynomial that we want to find with a leading coefficient is equal to one. So for this to be a quartet polynomial, we can have up to four zeros. And if we want this to be a real valued or have real coefficients, this cortex polynomial, then we're gonna have to use the conjugate um complex zeros theorem, which tells us if we have a complex zero, which we can see that I and two ir zero for this function since F of X is equal to zero at those two X values. Um And if we have one complex um zero, then we also need to have the conjugate of that complex zero to have real valued coefficients for our core tech polynomial. So that would mean that F of negative I. Is equal to F of negative two I which is equal to zero. And so now we can actually set up this polynomial by um making it a product of factors where we have x minus I X plus I X minus two, I an x plus two I. And then let's multiply these together. So let's multiply these first two. We're going to get X squared and then we're going to get minus I squared. Since we're gonna get negative I. X. And positive I X. So those are going to cancel. So this is going to be X squared um minus negative or the square of negative one squared plus one. And then this is multiplied by X times x minus two, I times X plus two I is going to be X squared. And then we're going to have minus for I squared. So this is equal to X squared plus one, multiplied by x squared plus four. And so now we can do this last multiplication here. So we're gonna get X to the fourth, going to get plus four X squared, and then we're going to get plus X squared and plus four. So this is equal to X the fourth Plus five x squared plus four. And we can see that our lives are leading coefficient A here was supposed to equal one, and we can see that it is equal to one here, since we just have this X to the fourth. So this is a core check polynomial with the given conditions on it um that we found just by using the complex con gets zeros there.

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