TITLE: O level Expansion problem.
QUESTION [2 upvotes]: yep, title sucks..

Q: The expansion of (1+px+qx2)8 = 1+8x+52x2+kx3.
   Find the values of p, q and k.

I found the values of $p$ and $q$ and they are:

p = 1
q = 3

But I am unable to find the value of $k$.

ANS: k = 224.

REPLY [1 votes]: The expansion up to the $x^3$ term gives us the following $$1+8px+28p^2x^2+8qx^2+56p^3x^3+56pqx^3.$$
Now, comparing coefficients with the expression on the right hand side gives $$8p=8,\quad 28+8q=52,$$ which gives $p=1$ and $q=3$. Then, substituting these values to find $k$, we get $$k=56+168=224,$$ as desired.
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