Video Transcript
Consider the expansion of 𝑥 to the
fifth power over eight minus eight over 𝑥 all raised to the ninth power in
descending powers of 𝑥. For which values of 𝑥 is the sum
of the middle two terms equal to zero?
In this question, we have a
binomial expression written in the form 𝑎 plus 𝑏 raised to the power of 𝑛. When expanding any expression of
this type, we know it will have 𝑛 plus one terms. This means that our expansion will
have 10 terms, and the middle two are the fifth and the sixth. We’re interested in 𝑎 sub five and
𝑎 sub six, the fifth and sixth terms of the expansion.
We know the general term of any
binomial expansion 𝑎 sub 𝑟 plus one is equal to 𝑛 choose 𝑟 multiplied by 𝑎 to
the power of 𝑛 minus 𝑟 multiplied but 𝑏 to the power of 𝑟. The fifth term is therefore equal
to nine choose four multiplied by 𝑥 to the fifth power over eight to the fifth
power multiplied by negative eight over 𝑥 to the fourth power.
The sixth term, on the other hand,
is equal to nine choose five multiplied by 𝑥 to the fifth power over eight to the
fourth power multiplied by negative eight over 𝑥 to the fifth power. We are told that the sum of these
two terms is equal to zero. This means that the fifth term is
equal to the negative of the sixth term. We notice that nine choose four is
equal to nine choose five as they are both equal to nine factorial divided by five
factorial multiplied by four factorial.
We can therefore cancel this on
both sides of our equation. Both sides of the equation also can
be divided by 𝑥 to the fifth power over eight raised to the fourth power. This means that the left-hand side
becomes 𝑥 to the fifth power over eight. We can also divide both sides by
negative eight over 𝑥 to the fourth power. The right-hand side becomes
negative negative eight over 𝑥. The two negatives become a
positive.
We can then cross multiply. We can multiply both sides by eight
and 𝑥. This gives us 𝑥 to the sixth power
is equal to 64. We can then take the sixth root of
both sides. This gives us 𝑥 is equal to
positive or negative two as positive and negative two both raised to the power of
six give us 64.
The word “values” in the question
suggests we will have more than one answer. The sum of the middle two terms is
equal to zero when 𝑥 is equal to negative two or two.
