Text description - screencast for exercise 5

[The narrator reads out the onscreen text.]

NARRATOR: 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 plus 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 equal 2x, b equal negative 3y and n equal 4, and substituting these expressions into the binomial theorem gives the following:

[(2x minus 3y) all to the power of 4 equals the sum of r equal to 0 up to r equal to 4 of 4 choose r times (2x) to the power of (4 minus r) times (-3y) to the power of r.]

NARRATOR: 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) plus (216 x squared y squared) minus (216 xy cubed) and plus (81y to the power of 4).

Part b. Again we use the binomial theorem and in this case we let a equal x and b equal 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 plus 24 minus 32 divided by x squared plus 16 divided by x to the power of 4.