Find a polynomial function f with real coefficients having the given degree and zeros. Answers w

Find a polynomial function f with real coefficients having...

Key Concepts

Polynomial Construction from Zeros

A polynomial can be constructed by forming factors corresponding to its zeros. Each zero a contributes a factor of (x - a), and the overall polynomial is obtained by multiplying these factors together, potentially adjusted by a nonzero leading coefficient. This technique bridges the gap between the algebraic property of roots and the analytical form of the polynomial.

Degree and Leading Coefficient Considerations

The degree of the polynomial is determined by the number of zeros (including their multiplicities) when expressed in factored form. Choosing an appropriate leading coefficient, which can be any nonzero real number, allows for flexibility in matching specific criteria. This concept ensures that the constructed polynomial not only has the desired zeros but also conforms to the prescribed degree.

Complex Conjugate Root Theorem

This theorem states that if a polynomial function has real coefficients and a non-real complex number is a zero, then its complex conjugate is also a zero. This concept is fundamental when constructing polynomials with real coefficients, as it ensures that non-real zeros always come in conjugate pairs, which when multiplied together yield quadratic factors with real coefficients.

Transcript

00:01 In this problem, you're going to be given a couple zeros of a polynomial function and ask to write the polynomial function.

00:10 Before we start the problem, i want to go over with a very important concept, and that is that if x1 is a zero of a polynomial function, then x minus x1 is a factor of the function.

00:28 Very important idea.

00:29 Also a very important idea of this that we're using this problem is something called the conjugate pair theorem.

00:43 And what this tells you is that if one complex number is a solution or a zero of a polynomial function, then its conjugate is also.

00:57 Okay, so with that in mind, let's start this problem.

01:01 We are given the fact that there is a polynomial function.

01:04 Function of degree four and it has two zeros that are given to us of i and one plus two i and we are derived the polynomial function with these conditions well the conjugate pair theorem tells us that if i is a zero then it's conjugate negative i is and if one plus two i is a zero so is it conjugate one minus two i so those are the four zeros of this polynomial function and it can only have at most four zeros okay so now we're ready to try writing the polynomial function so i'm going to say that the function is some wheel number a times x minus i times x minus i times x minus a negative i so i become x plus i times x minus 1 plus 2 i times x minus 1 minus 2 i now we need to take this and put it in a standard form so i'm going to do a couple different things on this step i'm going to go ahead and multiply the two princes of the front together so that's x square minus i square and i'm going to do a little bit of rearranging distributing the subtraction sign through the parentheses on the third and the fourth parentheses and then this is x minus 1 plus 2 i now keep in mind that i square is equal to a negative 1 so any time we have an i square we can replace it with a negative 1 okay so this first princess can become oops this first princess can become x squared plus 1 now i'm going to do something a little different on the second parentheses.

03:23 I'm going to write this as x minus 1 minus 2 i and you see y in just a minute and a similar type thing on the last parentheses...

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