forming polynomials help

[hotpotato1092]

I have no idea how to form polynomials when given zeros, and I have to for my homework. Here are a couple of the ones I have to do, can someone explain how to do them and give me the answer? The question states: Form a polynomial f(x) with real coefficients having the given degree and zeros. (Hint: Simplify so that there are no i's in your polynomial.)

1. Degree 4; zeros: 3 + 2i; 4, multiplicity 2

2. Degree 5; zeros: 2, multiplicity 1; -i; 1 + i

3. Degree 4; zeros: 3, multiplicity 2; -i

I'd really appreciate the help, thanks!

 

[srmichael]

Now, now. I'm not gonna give you the answers, but I will give you a good hint. Basically you are being asked to create a polynomial given the roots. Normally it's reversed. Usually you are asked to find the roots of a polynomial. To find the polynomial given the roots you would take the product of x minus each root. So if you have roots 1 and 2, then the polynomial would be $\displaystyle (x - 1)(x - 2)$. You just multiply it out and combine like terms.

Now, if you have complex numbers then the roots are always in conjugate pairs. So if you have 2 + i as a root the other root is 2 - i. The the polynomial would be $\displaystyle f(x) = [x - (2 + i)][x - (2 - i)]$. The same goes for irrational roots. They come in conjugate pairs as well. So if you are given one root is $\displaystyle -3 + \sqrt{2}$ then you would have to know the other root would be $\displaystyle -3 - \sqrt{2}$.

Oh yeah, remember $\displaystyle i^2 = -1$

If you are given multiplicities, then you take the (x - root) term and raise it to multiplicity value. So a polynomial with a root of 3 with multiplicity of 4 and a root of -5 with multiplicity of 2 would be set up as $\displaystyle f(x) = (x - 3)^4(x + 5)^2$.

Hope this helps!!