Finding a polynomial function with given zeroes

• Feb 25th 2013, 08:17 PM
zsf1990
Finding a polynomial function with given zeroes
To be exact, "Find a polynomial function with real coefficients that has the given zeros." My instructor just glazed over this in lecture and I don't see how the book is getting its answers. What exactly is the process to doing this? An example set of given zeros:

$\displaystyle 4, -3i$
• Feb 25th 2013, 08:41 PM
Prove It
Re: Finding a polynomial function with given zeroes
Quote:

Originally Posted by zsf1990
To be exact, "Find a polynomial function with real coefficients that has the given zeros." My instructor just glazed over this in lecture and I don't see how the book is getting its answers. What exactly is the process to doing this? An example set of given zeros:

$\displaystyle 4, -3i$

For your polynomial function to have real coefficients, all nonreal solutions will occur as complex conjugates. So you know that x = 4, x = -3i and x = 3i will all be zeroes of the polynomial. That means a polynomial that works is

\displaystyle \displaystyle \begin{align*} y &= a\left( x - 4 \right) \left[ x - (-3i) \right] \left( x - 3i \right) \\ &= a \left( x - 4 \right) \left( x^2 + 9 \right) \end{align*}