# Binomial Expansion

1. Homework Statement

Expand (1-6x)^4 (1+2x)^7 in ascending powers of x up to and including the terms in x^3

2. Homework Equations

(1-6x)^4 (1+2x)^7

3. The Attempt at a Solution

Firstly, I expand (1-6x)^4
= (1)^4 + 4C1 (-6x) + 4C2 (-6x)^2 + 4C3 (-6x)^3 + ...
= 1 + 4(-6) + 6 (36x^2) + 4(-216^3) + ...
= 1 -24x + 216^2 - 864x^3 + ...

Then after that, I expand (1+2x)^7
= (1)^7 + 7C1 (2x) + 7C2 (2x)^2 + 7C3 (2x)^3 +...
= 1 + 7 (2x) + 21 (4x^2) + 35 (8x^3) + ...
= 1 + 14x + 84x^2 + 280x^3 + ...

Finally, I expand (1-6x)^4 (1+2x)^7 together...
= (1-24x+216x^2=864x^3+...) (1+14x+84x^2+280x^3+...)
= 1+14x+84x^2+280x^3-24x-336x^2-2016x^3+3024x^3-864x^3+...
= 1 -10x-252x^2+2240x^3+...

My final answer is wrong. It should be 1-10x-36x^2+424x^3+...

Can someone help me solve this? Thanks.:tongue:

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AlephZero
Homework Helper
= (1-24x+216x^2=864x^3+...) (1+14x+84x^2+280x^3+...)

That's OK

= 1+14x+84x^2+280x^3 OK
-24x-336x^2-2016x^3 OK
+3024x^3-864x^3+... That's where you went wrong. There's a term missing, and probably you made another mistake adding up the terms because I can't see how you got 2240x^3 from your (wrong) numbers.