The cubic pattern

In case you're interested (and someone asked for this, so I know you might be!), here's the pattern we discovered today with the cubic polynomial.

Start with a function.

f(x) = 

Find the zeros.  We picked out 2 and 4 (checked with synthetic division) and ended up with 3/5 as our last.

How can you build the cubic polynomial if you know the zeros (without doing all of the messy binomial multiplication)?  Here we go!

Step 1:  Factor out the 5.  (This only works if the leading coefficient of 1).



Step 2.  Find the product of the zeros. 
                  (Which is the opposite of the constant term.)

Step 3.  Find the sum of the zeros. 
                   (Which is the opposite of the quadratic term.)

Step 4.  Find the sum of the pairwise products which is .
  (Which is the same as the linear term.)

So if you have the zeros of a cubic polynomial with leading coefficient of 1, you can use these patterns to find the polynomial itself.  I thought that was pretty cool! (Could you tell?  :) )

The next problem was to find a cubic polynomial with zeros at 5, . (It ends up being f(x) =  if you're playing along at home.)