we want expand the quantity 5x -2
raised the fourth power using the binomial theorem as well as Pascal's triangle
if we take a look at the binomial theorem it can be expressed using
summation notation
or an expanded notation as we see here notice if we take a look at
each term in the expansion each term has a factor has a combination
highlighted here in pink as well as a factor
involving x, the first term in the binomial
and a factor y from the second term at the binomial
as we look at our example you'll see the pattern in this
that make it much easier to remember in this video to buy with the combinations
instead of using factorials as we did in some other previous examples
we will use Pascal's triangle given here below
where if the binomial was raised to the nth power if we find row n
in Pascal's triangle it'll give us the value of each
combination if you are no more information about how this triangle is
formed
I'll put a link on the screen for a video that will go over this in more detail
so they go back to our example notice our binomial is raised to the fourth
power
which means our first term will have the combination four choose zero
and then because the first term in the binomial is 5x will have four factors of
5x in the second term is -2
so we'll have 0 factors -2 so we started with four choose zero
now we have four factors a 5x and 0 factors have -2
and then for the next term will have four choose one
and now we only have three factors 5x
so we have 5x to the third and we start picking up backers factors of -2
so we'll have negative two to the first then for the third term will have four choose two
and then we'll have 5 x to the second
and -2 to the second
notice if we add these exponents each time they sum to
four which is the orignial exponent of the binomial
next time we'll have four choose 3
ane we'll only have one factor
5x and three factors have negative two
and now for the last term will have four choose four
and we'll have 0 factors 5x
and four factors have negative two
so we've now applied the binomial theorem
so the next step is to simplify each term in the expansion
and as mentioned earlier to evaluate the combinations
we're gonna use Pascal's triangle
and since the original binomial is raised to the fourth power
we can find the value of each combination in row four
of Pascal's triangle where the first row here is row zero
so we have row zero row one
row two row three and finally row four
another way to check to make sure that we have the correct row
is to see that this role has five numbers and our expansion has
5 combinations therefore four choose zero is equal to one
for choose one is equal to four four choose two is equal to 6
and so on so again four choose zero is equal to one so we'll have one
times five x to the fourth well five to the fourth is equal to 625
so we'll have 625 x to the fourth
-2 to the zero is equal to one
plus four choose one which is equal to 4
so we have four times 5x
to the third well, five in the third is equal to 125
so we have 125 X to the third times -2
plus four choose two is equal to 6
so at six times 5 x square that would be 25 x squared
-2 squared is positive
four plus
four choose three is equal to 4 so we have four times
5x times negative two to the third that's negative
8 and finally our last term we have four choose four that's one
5x to the zero 0 that's one
and then we have negative two to the fourth which should be
positive 16 so we'll simplify this one more time
we have 625 X in the fourth
the next term we have four times 125
times -2 that's actually -1000
times X to the third so we have minus 1000x
the third next we have six times twenty five times four
that's 600 times x to the second so plus
600 X in the second
next we have four times five times -8
that's -160 times X so minu 160 X
and finally we have one times one time sixteen so we have plus 16
this would be the expansion the quantity 5x -2
raised to the fourth
hopefully we agree if we recognize the pattern in the binomial theorem
it makes it a lot easier to remember and apply the theorem
to expand binomails raised to powers i hope you this found this helpful