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# Factoring difference of squares: leading coefficient ≠ 1

Video transcript

Let's see if we can
factor the expression 45x squared minus 125. So whenever I see
something like this-- I have a second-degree
term here, I have a subtraction
sign-- my temptation is to look at this as a
difference of squares. We've already seen
this multiple times. We've already seen that if
we have something of the form a squared minus b squared, that
this can be factored as a plus b times a minus b. So let's look over here. Well, over here,
it's not obvious that this right over
here is a perfect square. Neither is it obvious
that this right over here is a perfect square. So it's not clear
to me that this is a difference of squares. But what is interesting
is that both 45 and 125 have some factors in common. And the one that
jumps out at me is 5. So let's see if we can factor
out a 5, and by doing that, whether we can get something
that's a little bit closer to this pattern right over here. So if we factor out a 5,
this becomes 5 times-- well, 45x squared divided by 5
is going to be 9x squared. And then 125 divided by 5 is 25. Now, this is interesting. 9x squared-- that's
a perfect square. If we call this a
squared, then that tells us that a
would be equal to 3x. 3x-- the whole thing
squared is 9x squared. Similarly-- I can never say
similarly correctly-- 25 is clearly just 5 squared. So in this case, if we're
looking at this template, b would be equal to 5. So now this is a
difference of squares, and we can factor it completely. So we can't forget our 5 out
front that we factored out. So it's going to be
5 times a plus b. So let me write this. So it's going to be 5 times
a plus b times a minus b. So let me write the b's
down, plus b and minus b. And we're done. 5 times 3x plus 5
times 3x minus 5 is 45x squared minus
125 factored out.