1. ## Factoring Polynomial

I have an old problem that I forgot how I got the answer. The problem is: Find the factored form of a polynomial with real coefficients f(x) that is degree 4, zero's @ 2i and 3 (multiplicity of 2). The function must satisfy f(0) = 72.

2. Hello,
Originally Posted by galanm
I have an old problem that I forgot how I got the answer. The problem is: Find the factored form of a polynomial with real coefficients f(x) that is degree 4, zero's @ 2i and 3 (multiplicity of 2). The function must satisfy f(0) = 72.
If it's degree 4, then it can be written in this form : $\displaystyle f(x)=ax^4+bx^3+cx^2+dx+e$

Then, you have $\displaystyle f(2i)=0 ~,~ f(3)=0 ~,~ f(0)=72$
This gives you 3 equations.

Since 3 is a zero with multiplicity of 2, this means that $\displaystyle f'(3)=0$ as well

Now, you only have 4 equations and you need a fifth ?
Since f is a polynomial with real coefficients, if a complex number is a zero, then its conjugate is also a zero.
Thus $\displaystyle f(-2i)=0$

Looks good to you ?

3. What I have is (x+2i)(x-2i)(x-3)(x-3) the instructor marked it wrong and wrote in a 2 in front of the first term.

4. ## I see what I did

zeros @2i, -2i (conjugate), 3, and 3
(x+2i)(x-2i)(x-3)(x-3)
This is what I forgot--SUBSTITUTE 0 for x's gives us 36. Which means we had to multiply the string gy 2 to satisfy f(0)=72.

Thanks for the sounding board.