find the coefficient of the term when the expression is expanded.

Question

$a^2x^3$: $(a + x + c)^2(a + x + d)^3$

I am considering to list all cases of $a^nx^m$ from both expansion that sum of n=2, sum of m=3. Like in the first case: $a^0x^0$ from first expansion and $a^2x^3$ from the second expansion.

Then calculate the coefficient of $a^2x^3$ in each case by trinomial expansion and sum them up. But this way is way to complex. I am wondering if there is any idea to make it easier?

Thank you!

Answer 1

It is convenient to use the coefficient of operator $[x^n]$ to denote the coefficient of $x^n$ of a series. This way we can focus at the terms which contribute to $[a^2x^3]$ and can skip other terms.

We obtain \begin{align*} \color{blue}{[a^2x^3]}&\color{blue}{(a+x+c)^2(a+x+d)^3}\\ &=[a^2x^3](a+(x+c))^2(a+(x+d))^3\\ &=[a^2x^3]\left(a^2+2a(x+c)+(x+c)^2\right)\\ &\qquad\qquad\quad\cdot\left(a^3+3a^2(x+d)+3a(x+d)^2+(x+d)^3\right)\\ &=[a^2x^3]\left(a^2(x+d)^3+2a(x+c)3a(x+d)^2+(x+c)^23a^2(x+d)\right)\tag{1}\\ &=[a^2x^3]\left(a^2x^3+6a^2x^3+3a^2x^3\right)\tag{2}\\ &\,\,\color{blue}{=10} \end{align*}

Comment:

• In (1) we multiply out leaving only terms which contribute to $[a^2x^3]$.

• In (2) we multiply out once more again leaving only terms which contribute to $[a^2x^3]$.