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## Factoring polynomials with special product forms

# Factoring using the perfect square pattern

CCSS.Math:

## Video transcript

We need to factor 25x to
the fourth minus 30x squared plus 9. And this looks really daunting
because we have something to the fourth power here. And then the middle term
is to the second power. But there's something
about this that might pop out at you. And the thing that pops out at
me at least is that 25 is a perfect square, x to the fourth
is a perfect square, so 25x to the fourth is
a perfect square. And 9 is also perfect square, so
maybe this is the square of some binomial. And to confirm it, this center
term has to be two times the product of the terms that you're
squaring on either end. Let me explain that a
little bit better So, 25x to the fourth, that is
the same thing as 5x squared squared, right? So it's a perfect square. 9 is the exact same thing as,
well, it could be plus or minus 3 squared. It could be either one. Now, what is 30x squared? What happens if we take 5
times plus or minus 3? So remember, this needs to be
two times the product of what's inside the square, or
the square root of this and the square root of that. Given that there's a negative
sign here and 5 is positive, we want to take the
negative 3, right? That's the only way we're going
to get a negative over there, so let's just try
it with negative 3. So what is what is 2 times 5x
squared times negative 3? What is this? Well, 2 times 5x squared is 10x
squared times negative 3. It is equal to negative
30x squared. We know that this is
a perfect square. So we can just rewrite this as
this is equal to 5x squared-- let me do it in the
same color. 5x squared minus 3 times
5x squared minus 3. And we saw in the last
video why this works. And if you want to verify
it for yourself, multiply this out. You will get 25x to the fourth
minus 30x squared plus 9.