SOLUTION: Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros. 5-i, SQUARE ROOT 11

Question 1181514: Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros. 5-i, SQUARE ROOT 11
Answer by Solver92311(821) (Show Source): You can put this solution on YOUR website!

Both complex and irrational zeros come in conjugate pairs. This means that if

is a zero, then so is

. And if

where

and

then

is also a zero.

Since the degree of a polynomial function is equal to the number of zeros, and since you are given one complex zero and one irrational zero there must, at a minimum, be four zeros, you are looking for a fourth-degree polynomial.

We also know that if

is a zero of a polynomial function, then

must be a factor of the polynomial.

You were given

as a zero, so

is a factor of the desired polynomial. Likewise,

is a factor. Also, you were given

as a zero so

is a factor of the desired polynomial and likewise

is a factor.

Putting it all together, your polynomial function, in factored form, is:

The only thing left for you to do is to multiply the four binomials and collect like terms. Hints: The product of two conjugates is the difference of two squares and

.

John

My calculator said it, I believe it, that settles it

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