Video Transcript
Find the coefficient of 𝑥 cubed in
the expansion of two plus three 𝑥 all to the eighth power.
This is an example of a binomial
expression written in the form 𝑎 plus 𝑏 all raised to the 𝑛th power. We can solve problems of this type
using Pascal’s triangle. However, as the power or exponent
in this case is larger than five, it would be very time-consuming to write out every
row of Pascal’s triangle. We will therefore use the binomial
theorem. This states that 𝑎 plus 𝑏 to the
𝑛th power is equal to 𝑛 choose zero multiplied by 𝑎 to the 𝑛th power plus 𝑛
choose one multiplied by 𝑎 to the 𝑛 minus oneth power multiplied by 𝑏 to the
first power, and so on, where 𝑛 choose 𝑟 is equal to 𝑛 factorial divided by 𝑛
minus 𝑟 factorial multiplied by 𝑟 factorial. As we move along term by term, the
powers or exponents of 𝑛 decrease, whereas the powers or exponents of 𝑏
increase.
In this question, the value of 𝑎
is two, 𝑏 is equal to three 𝑥, and the power or exponent 𝑛 equals eight. At this stage, we could write out
the whole expansion. However, we are only interested in
the term containing 𝑥 cubed. As 𝑏 is equal to three 𝑥, this
will be the term in the general expansion containing 𝑏 cubed. This term is equal to eight choose
three multiplied by 𝑎 to the fifth power multiplied by 𝑏 cubed.
Substituting in our values of 𝑎
and 𝑏, we have eight choose three multiplied by two to the fifth power multiplied
by three 𝑥 all cubed. Eight choose three is equal to
eight factorial divided by five factorial multiplied by three factorial. We recall that eight factorial can
be rewritten as eight multiplied by seven multiplied by six multiplied by five
factorial. We can then cancel five factorial
from the numerator and denominator. Three factorial is equal to
six. We can therefore cancel this from
the numerator and denominator. Eight choose three is therefore
equal to eight multiplied by seven, which is equal to 56.
Alternatively, we could’ve just
typed this straight in to our calculator. Two to the fifth power is equal to
32. As three cubed is equal to 27,
three 𝑥 all cubed is 27𝑥 cubed. Our expression becomes 56
multiplied by 32 multiplied by 27𝑥 cubed. This is equal to 48,384𝑥
cubed. As we just want the coefficient of
𝑥 cubed, the final answer is 48,384.
