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.
2026-03-26 22:12:00.1774563120
O level Expansion problem.
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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.