We want to expand the quantity 2x minus 3y squared
raised to the fourth power, using The Binomial Theorem,
and this video will also by using Pascal's Triangle.
Here's The Binomial Theorem stated using Summation Notation
as well as in Expanded Form.
Notice in expanded form, each term has three factors,
where one factor consists of a combination.
Each term contains a factor of X,
which comes from the first term in the binomial.
And each term also contains the factor of Y
from the second term in the binomial.
So while this looks fairly involved,
there is a nice pattern to make the Binomial Theorem
easy to remember and apply.
And in this video, to evaluate these combinations,
we're going to use Pascal's Triangle
where if we have the binomial raised to the nth power,
if we find row n in Pascal's Triangle,
it'll give us the value of each of these combinations.
If you're wondering how to form Pascal's Triangle,
we start with a triangle of three ones as we see here
and notice each row begins and ends with a one.
And if the number's inbetween,
we have the two numbers above.
Notice that one plus one is equal to two.
One plus two is equal to three.
Two plus one is equal to three.
One plus three is equal to four.
Three plus three is equal to six.
Three plus one is equal to four. And so on.
So going back to our example,
notice the binomial is raised to the fourth power,
which means the first term will have a factor of
four choose zero.
And then because the first term is 2x,
we'll have four factors of 2x.
And because the second term would be negative 3y squared,
we'll have zero factors of negative 3y squared.
So this would be the first term in the binomial expansion
and for the next term, this is four choose zero,
we'll have four choose one.
If we have four factors of 2x here,
we'll have one less, so we'll have 2x to the third.
And if we have zero factors of negative 3y squared,
we'll have one factor of negative 3y to the second.
So for each successive term,
we're losing a factor of 2x
and gaining a factor of negative 3y squared.
Plus the third term will have four choose two.
So we'll only have two factors of 2x
or 2x to the second
and we'll have two factors of negative 3y squared.
The fourth term will have four choose three.
And we'll only have one factor of 2x,
so 2x to the first.
And we'll have three factors of negative 3y to the second.
So this raised to the third power.
And for our last term,
we'll have four choose four,
zero factors of 2x,
and four factors of negative 3y to the second.
Just applied the Binomial Theorem
and now need to simplify to find each term.
And again for this example,
we're going to use Pascal's Triangle
to evaluate our combinations.
And since the original binomial
is raised to the fourth power,
we can find the value of each combination here
in row four of Pascal's Triangle,
starting with row zero, so you have row zero,
row one, row two, row three, and row four.
Notice how there are five combinations
and five numbers in this row.
So four choose zero is equal to one.
Four choose one is equal to four.
Four choose two is equal to six and so on.
So for our first term, four choose zero would be one
times 2x raised to the fourth.
Well two to the fourth is equal to 16,
so we'll have 16x to the fourth,
negative 3y to the second raised to the zero power
is equal to one
plus four choose one
is equal to four,
times 2x to the third.
Two to the third is equal to eight, we have 8x cubed.
And then we have one factor of negative 3y squared,
plus four choose two, that's equal to six,
times 2x squared, that'll be 4x squared.
And then we have negative 3y to the second squared,
that'll be positive 9y to the fourth.
Next, we have four choose three that's equal to four.
2x to the first, that's just 2x,
times negative 3y to the second cubed.
Well, negative 3 to the third would be negative 27,
and then y squared to the third would be y to the sixth.
And for the last term, we have four choose four
that's equal to one as we see here.
2x to the zero is equal to one.
And then negative 3y squared to the fourth,
negative three to the fourth is equal to 81.
Y squared to the fourth is equal to y to the eighth.
So we need to simplify this one more time
to find our expansion.
So, first term we can see would just be 16x to the fourth.
For the next term, the coefficient would be
four times eight times negative three,
that's negative 96.
And then we have x to the third, y to the second.
So minus 96x to the third y to the second.
For the next term, the coefficient would be
six times four times nine.
That's equal to 216
and then we'll have x to the second, y to the fourth.
So, plus 216x to the second,
y to the fourth.
The next term would have a coefficient
of four times two times negative 27.
That's negative 216, so we have minus 216
and we have one factor of x and six factors of y.
So we have xy to the sixth.
And our last term, we can see it's just going
to be plus 81y to the eighth.
Okay, that's going to do for this example.
I hope you found this helpful.