# How do you factor given that #f(9)=0# and #f(x)=x^3-18x^2+95x-126#?

##### 1 Answer

The equation becomes:

#### Explanation:

Hmm, I don't know... Let's try it out, shall we?

First, you're given that

and

Well, let's check if that's true!

i.e.

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 give

This also means that "if" 9 is really a solution to our equation, we can factor it out, and the equation should look like:

where

Now, let's open the parentheses and let's gather all similar terms together:

i.e.

Great!

Now, we can relate this to our first equation.

This means that

Let's solve for

Moving to the second line, we then have

i.e.

and finally,

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:

Q.E.D.