WOMAN: Exercise 5.
Expand
A, (2x minus 3y) all to the power of 4,
B, (x minus 2 divided by x) all to the power of 4.
Part A.
We use the binomial theorem,
which states that (a + b) all to the power of n
is equal to the sum of r equal to 0 up to r equal to n
of n choose r times a to the power of n minus r
times b to the power of r.
In this case, we wish to let a = 2x,
b = negative 3y
and substituting these expressions into the binomial theorem
gives the following:
And expanding the sum gives the following:
And then simplifying and simplifying again gives the final expansion,
which says that (2x – 3y) all to the power of 4
is equal to 16x to the power of 4
minus 96x cubed y
+ 216 x squared y squared
minus 216 xy cubed
and + 81y to the power of 4.
Part B.
Again we use the binomial theorem and in this case we let a = x
and b = negative 2 divided by x and, again, n will equal 4.
Substituting these expressions into the binomial theorem
and then expanding the sum
and simplifying to give the final expansion
which says that x minus 2 divided by x all to the power of 4
is equal to x to the power of 4 minus 8x squared + 24
minus 32 divided by x squared
+ 16 divided by x to the power of 4.