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# Special products of the form (ax+b)(ax-b)

Video transcript

Find the product 2x
plus 8 times 2x minus 8. So we're multiplying
two binomials. So you could use FOIL,
you could just straight up use the distributive
property here. But the whole point of
this problem, I'm guessing, is to see whether you
recognize a pattern here. This is of the form a
plus b times a minus b, where here a
is 2x and b is 8. We have 2x plus 8 and then 2x
minus 8. a plus b, a minus b. What I want to do
is I'm just going to multiply this out for us. And then just see what happens. Whenever you have this pattern,
what the product actually looks like. So if you were to
multiply this out, we can distribute the a plus b. We could distribute
this whole thing. Distribute the whole
a plus b on the a and then distribute it on the b. And I could have done this
with this problem right here, and it would have taken us
less time to just solve it. But I want to find out
the general pattern here. So a plus b times a. So we have a times a plus
b, that's this times this. And then a plus b
times negative b, that's negative
b times a plus b. So I've done distributive
property once, now I could do it again. I can distribute the a
onto the a and this b and it gives me a squared. a times a is a squared,
plus a times b, which is ab. And now I can do it
with the negative b. Negative b times a is negative
ab or negative ba, same thing. And negative b times b
is negative b squared. Now, what does this simplify to? Well, I have an ab, and
then I'm subtracting an ab. So these two guys
cancel out and I am just left with a squared
minus b squared. So the general pattern,
and this is a good one to just kind of
know super fast, is that a plus b times
a minus b is always going to be a squared
minus b squared. So we have an a plus
b times an a minus b. So this product is
going to be a squared. So it's going to be
2x squared minus b squared minus 8 squared. 2x squared, that's
the same thing as 2 squared times x
squared, or 4x squared. And from that, we're
subtracting 8 squared. So it's going to be
4x squared minus 64.