Write a polynomial of the smallest degree with roots -8, 9, 10, and -11.

Question

solomonzelman · Answer

The smallest polynomial with roots:
$\large\color{black}{ \displaystyle \color{blueviolet}{\rm a}, \color{blue}{~\rm b},\color{brown}{~\rm c},\color{green}{~\rm d} }$

$\large\color{black}{ \displaystyle f(x)=(x-\color{blueviolet}{\rm a})(x-\color{blue}{\rm b})(x-\color{brown}{\rm c})(x-\color{green}{\rm d}) }$

solomonzelman · Answer

This is also true with more roots or less roots.
For example, below:

The smallest polynomial with roots:
$\large\color{black}{ \displaystyle \color{blueviolet}{\rm a}, \color{blue}{~\rm b},\color{brown}{~\rm c},\color{green}{~\rm d},\color{darkgoldenrod}{~\rm k},\color{magenta}{~\rm l} }$

$\large\color{black}{ \displaystyle f(x)=(x-\color{blueviolet}{\rm a})(x-\color{blue}{\rm b})(x-\color{brown}{\rm c})(x-\color{green}{\rm d})(x-\color{darkgoldenrod}{\rm k})(x-\color{magenta}{\rm l}) }$

solomonzelman · Answer

Should I give you an example with numbers?

anonymous · Answer

No, I think I get it now. Thanks

solomonzelman · Answer

Ok, do you want to post your answer for me to make sure it is correct?

anonymous · Answer

Sure, I am getting x^4-183x^2+38x+7920 as my answer

solomonzelman · Answer

* Yes, that is exactly right! *

anonymous · Answer

Cool!! thanks!

solomonzelman · Answer

You welcome!