Exercise 2 requires us to expand and simplify
the following sets of brackets.
In order to do this,
we'll need to multiply each term in the second bracket by x, et cetera .
And then each term in the second bracket by –1, et cetera.
So x multiplied by the second bracket
and then –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-1, x to the power of n-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-1
leaves us with x to the power of 1 plus n-1, which is n.
Similarly, x to the 1 times x to the n-2
leaves us with x to the n-1.
The next term would give us x to the n-2, et cetera,
down to x cubed ...
... x squared
and x.
Then multiplying the second bracket by –1
simply makes each term negative, so we get -x to the n-1,
-x to the n-2 ... minus ...et cetera,
-x squared -x –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 +x to the power of n-1
and then -x to the power of n-1, which will cancel out.
Similarly, +x to the n-2, -x to the n-2.
+x cubed, and there would be a -x cubed here.
+x squared, -x squared.
+x, -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-1.