# Thread: Trig~Polynomial and Rational fuctions..Please help me

1. ## Trig~Polynomial and Rational fuctions..Please help me

Well I need this by the end of tonight. I still haven't figured out the answer. Could someone please help, thanks.

Form a polynomial F(x) with real coefficients having given degree and zero.

Degree 4; zeros: i, 1+ 2i

Thanks, this will be greatly appreciated.

2. Originally Posted by nib-ball
Well I need this by the end of tonight. I still haven't figured out the answer. Could someone please help, thanks.

Form a polynomial F(x) with real coefficients having given degree and zero.

Degree 4; zeros: i, 1+ 2i

Thanks, this will be greatly appreciated.
there should be four zeroes. if any of the zeroes have multiplicity, please identify them

3. i need the exact help with this.. precalc finals tomorrow. and im clueless on 50% of the stuff.

4. Hello, nib-ball!

Form a polynomial $\displaystyle f(x)$ with real coefficients
of degree 4, and zeros: $\displaystyle i,\:1+ 2i$
You're expected to know a few facts about polynomial equations . . .

Complex zeros always appear in conjugate pairs.
. . Since $\displaystyle i$ is a zero, then $\displaystyle -i$ is also a zero.
. . Since $\displaystyle 1 + 2i$ is a zero, then $\displaystyle 1 - 2i$ is also a zero.

If $\displaystyle a$ is a zero of a polynomial, then $\displaystyle (x - a)$ is a factor of the polynomial.
. . Since $\displaystyle i$ is a zero, then $\displaystyle (x - i)$ is a factor.
. . Since $\displaystyle -i$ is a zero, then $\displaystyle (x + i)$ is a factor.
. . Since $\displaystyle 1 + 2i$ is a zero, then $\displaystyle \left(x - [1 + 2i]\right)$ is a factor.
. . Since $\displaystyle 1 - 2i$ is a zero, then $\displaystyle (x - [1 - 2i])$ is a factor.

The polynomial could be: .$\displaystyle f(x) \;=\;(x - i)(x + i)(x - 1 - 2i)(x - 1 + 2i)$

. . which simplifies to: .$\displaystyle f(x)\;=\;x^4 - 2x^3 + 6x^2 - 2x + 5$