Question

Expand (4x - 3y)^5.

Answer 1

If you know how to expand (a+b)^5, you can also do (4x-3y)^5.
Knowledge of Pascal's Triangle comes in handy as well...

I think you know how to expand (a+b)^2:
\[(a+b)^2=(a+b)(a+b)=a^2+2ab+b^2\]

There are three terms with 1, 2 and 1 as coefficients.

If you would take the truble to expand (a+b)^3, you'd get:\[(a+b)^3=(a+b)(a+b)^2=(a+b)(a^2+2ab+b^2)=\]
\[....=a^3+3a^2b+3ab^2+b^3\]
Now there are FOUR terms, with coefficients 1, 3, 3, 1.

Answer 2

The next one, (a+b)^4:\[(a+b)^4=...=a^4+4a^3b+6a^2b^2+4ab^3+b^4\]
FIVE terms, with coefficients 1, 4, 6, 4, 1

Answer 3

If you think that (a+b)^5 has SIX terms, then you're right! ;)

We've got only two problems left now:

1. what are the terms?
2. what are their coefficients?

Answer 4

1, 5, 10, 10, 5, 1

Answer 5

Problem 1: what are the terms?

a^5 and b^5 you might have guessed. In fact the terms are:
\[a^5,a^4b,a^3b^2,a^2b^3,ab^4,b^5\]
See the pattern? Each term is the product of 5 numbers a and b. Each combination is possible, so that why we get these.

Answer 6

@ka4aka: you are right! So it is\[a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5\]
Now you "only" have to replace a with 4x and b with -3y and you're done!

Sadly, this is still a lot of work that can easily go wrong.
But this is all I can do for you ...

Answer 7

Thank you very much.