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.
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.
Hello,
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 ?