Expert: David Montiel - 6/27/2010

Question
Hello:

I want to determine the expanded form for the following:

P(1 + r)^3

I think that the answer is as follows:  (P + rP) + r(P + rP) + r((P + rP) + r(P + rP))

It has been a long time since I have had algebra. I am not a student. I am just interested in learning how this expanded version is determined.  I know this much: P(1 + r) = (P + rP).

This is determined by multiplying 1 + r by P. P(1 + r)(1 + r)(1 + r) should equal what is shown above. The multiplication becomes confusing to me after the second period is multiplied.
I do not know what to do after (P + rP) + r(P + rP).
I think that the final step is (P + rP) + r(P + rP)(1 +r).
But I do not know how to multiply at this point.

I thank you for your reply.

Answer
Hello Kenneth,

I'll be glad to help.

I think you are just a little confused; the answer you show is correct.

So, as you correctly point out:

P(1 + r)(1 + r) = (P + Pr)(1 + r) = [ (P + rP) + r(P + rP) ]

Now all we have to do is multiply the previous result by (1 + r) by carrying out the product of the term in square brackets "[]" by both "1" and "r" and add them up, which gives us

P(1 + r)(1 + r)(1 + r) = 1*[(P + rP) + r(P + rP)] + r*[(P + rP) + r(P + rP)]

which is what you have.

However I wouldn't call what you have the expanded form. I think you can go further and carry out the remaining products.

For instance

P(1 + r)(1 + r) = (P + rP) + r(P + rP) = (P + rP) + r(P + rP) = P + rP + rP + (r^2)P = P + 2Pr + Pr^2.

So, after a little work you should be able to get

P(1 + r)^3 = Pr^3 + 3Pr^2 + 3Pr + P,

which is a simpler way to express the same result you have.

Hope this helps,

David