### Video Transcript

Fully simplify π₯ minus three times
π₯ squared minus six π₯ plus nine over π₯ minus three cubed.

Remember, to simplify fractions, we
divide the numerator and denominator by the greatest common factor of each. Now, we can see that we have a
common factor of π₯ minus three. Thereβs an π₯ minus three in the
numerator and the denominator. But is that the greatest common
factor? Well, to check, we factor any
nonfactored expressions. So weβre going to fully factor the
expression π₯ squared minus six π₯ plus nine from the numerator of our fraction. This is a quadratic expression, but
there are no common factors in π₯ squared, negative six π₯, and nine. So that tells us we factor into two
parentheses.

We know that, in order to achieve
an π₯ squared, we need an π₯ here and an π₯ here. And we need to find two numbers
whose product is nine and whose sum is negative six. Well, that must be negative three
and negative three, since negative three times negative three is positive nine. But negative three plus negative
three is negative six. If we replace π₯ squared minus six
π₯ plus nine with its factored form, our fraction becomes π₯ minus three times π₯
minus three times π₯ minus three over π₯ minus three cubed. But of course, it should be quite
clear that π₯ minus three times π₯ minus three times π₯ minus three is actually π₯
minus three cubed.

Notice now that our numerator and
denominator are actually equal. Weβre dividing a number by
itself. And when we divide a number by
itself, we get one. So in simplified form, this
fraction is simply one. Now, note that we couldβve
approached this slightly differently. Instead of factoring at the start
π₯ squared minus six π₯ plus nine, we could have divided through by a factor of π₯
minus three. We eventually saw that this isnβt
the greatest common factor, but itβs a good start. We divide the numerator and the
denominator by π₯ minus three, noting that π₯ minus three cubed divided by π₯ minus
three is π₯ minus three squared.

Then we couldβve factored and
spotted that we had further common factors of π₯ minus three squared. Either method ultimately results in
us dividing both the numerator and denominator by the greatest common factor of π₯
minus three cubed. And either method results in an
answer of one.