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WOMAN: Exercise 8.

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Find the middle term of (2x minus 3y)
all to the power of 6.

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The expansion of (2x minus 3y) all
to the power of 6 has seven terms,

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so the middle term
is the fourth term.

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The general term in the expansion of
(2x minus 3y) all to the power of 6

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is 6 choose r
times 2x to the power of six minus r

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times negative 3y
all to the power of r,

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where r is between
0 and 6 inclusive.

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Given that the first term is the one
in which r is equal to 0,

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the fourth term is therefore the one
in which r is equal to 3,

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so the middle term
is the term where r = 3.

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So substituting in
we get six choose three

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times 2x all to the power of three

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times negative 3y
all to the power of 3,

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which simplifies to give

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20 times 8x cubed
times negative 27y cubed,

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which simplifies
to give a middle term

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of negative 4,320x cubed y cubed.