Text description - screencast for exercise 2

[The narrator reads out the onscreen text.]

NARRATOR: Exercise 2 requires us to expand and simplify the following sets of brackets.

[Equation: (x minus one) (x to the power of n minus one plus x to the power of n minus 2 plus ... plus x squared plus x plus 1.)]

NARRATOR: In order to do this, we'll need to multiply each term in the second bracket by x and then each term in the second bracket by negative 1, so x multiplied by the second bracket and then negative 1 multiplied by the second bracket.

So expanding these brackets out, in the first bracket we have x to the power of 1 multiplied by x to the power of n minus 1, x to the power of n minus 2, et cetera.

When multiplying terms with the same base, we know we can add the powers. So x to the power of 1 times x to the power of n minus 1 leaves us with x to the power of 1 plus n minus 1, which is n. Similarly, x to the 1 times x to the n minus 2 leaves us with x to the n minus 1. The next term would give us x to the n minus 2, et cetera, down to x cubed, x squared and x.

Then multiplying the second bracket by negative 1 simply makes each term negative, so we get negative x to the n minus 1, negative x to the n minus 2, minus et cetera, minus x squared minus x minus 1.

And so now we've expanded, we need to try to simplify and we can see there are a number of like terms. So we have plus x to the power of n minus 1 and then minus x to the power of n minus 1, which will cancel out. Similarly, plus x to the n minus 2, minus x to the n minus 2, plus x cubed, and there would be a minus x cubed here, plus x squared, minus x squared, plus x, minus x.

And once we're done with all the cancelling, all that we're actually left with is the first and last term of the expansion. So expanding out these brackets leaves us simply with x to the power of n minus 1.