Video Transcript
Find the third term in the
expansion of four 𝑥 plus three all cubed.
Well, if we actually think about
our expansion as 𝑎 plus 𝑏 to the power of 𝑛, then what we can actually think of
is actually we’ve got a general term formula. And it’s gonna actually help us
find any term of our expansion. And this general term formula tells
us that if we have a term 𝑟 plus one, then this is equal to 𝑛 choose 𝑟 multiplied
by 𝑎 to the power of 𝑛 minus 𝑟 multiplied by 𝑏 to the power of 𝑟.
Okay, so now we have our
formula. Let’s try and apply it to our
expansion to try find our third term. Well, we want to find the third
term. So therefore, we’re saying 𝑇
three. But this means that our 𝑟 is gonna
have to equal two.
So what about our 𝑛? What will our 𝑛 equal? Well, if we see that actually our
parentheses are raised to the power of three, then we know that 𝑛 is gonna be equal
to three. So we can now start writing out our
formula. So we can say that the third term
is equal to three choose two multiplied by our first term, which is four 𝑥, to the
power of three minus two, because that’s 𝑛 minus 𝑟, then multiplied by three to
the power of two again because three is our 𝑏 and two is our 𝑟.
Okay, great! So now let’s try and simplify. Well, first of all, we’re actually
gonna look at three choose two because we can actually find this on our
calculator. But what does this actually
mean? Well, what this actually means is
actually is the number of combinations of two items that can be selected of a set of
three items.
But how do we calculate it? Well, we calculate it using a
general formula, which is that 𝑛 factorial over 𝑟 factorial multiplied by 𝑛 minus
𝑟 factorial. So in this case, we have three
factorial over two factorial multiplied by three minus two factorial, which is just
gonna give us three multiplied by two multiplied by one over two multiplied by one
multiplied by one. And that’s because we find the
factorial by multiplying each of the integers greater than zero that are less than
or equal to the value that we have. So in this case, on the top here,
be three factorial would be three multiplied by two multiplied by one.
Okay, great! So this would just give us a result
of three because it’ll be three over one, which is just three. Okay, and if you check that on a
calculator, you’ll get the same value. Right now, we know what three
choose two is and where it comes from. Let’s get back on and find our
third term. So our third term is gonna be equal
to three multiplied by four 𝑥 multiplied by nine. So therefore, we can say that the
third term in the expansion of four 𝑥 plus three all cubed is gonna be equal to
108𝑥.
