How do you factor given that #f(9)=0# and #f(x)=x^3-18x^2+95x-126#?
The equation becomes:
Hmm, I don't know... Let's try it out, shall we?
First, you're given that
Well, let's check if that's true!
Since I'm too lazy to do the calculations,
let's say they are correct, and assume that
What's another equation that we know of, that also has 9 as its solution?
Can you think of a simple one?
This one for example:
This also means that "if" 9 is really a solution to our equation, we can factor it out, and the equation should look like:
Now, let's open the parentheses and let's gather all similar terms together:
Now, we can relate this to our first equation.
This means that
Let's solve for
Moving to the second line, we then have
So, our new equation looks like the following:
But let's not stop here! Maybe we can factor the right side one more time! For that, let's see if we can solve it if we set it to 0.
We solve it using the quadratic formula (see other posts!),
and we get:
which means we can rewrite the equation as such:
So, finally, the first equation becomes: