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# What is the coefficient of in

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What is the coefficient of  in

Guest Dec 4, 2014

#4
+20680
+10

What is the coefficient of  in

$$\\ \small{\text{ \begin{array}{l} \text{set }  1+x+x^2+x^3 = (1+x)(1+x^2)  \text{ we have }\left[(1+x)(1+x^2)+x^4 \right]^4 \\ \text{now we expand} \\ [(1+x)(1+x^2)]^4 \underbrace{ + 4*[(1+x)(1+x^2)]^3({x^4})+6*[(1+x)(1+x^2)]^2({x^4})^2+4*[(1+x)(1+x^2)]({x^4})^3+ ({x^4})^4 }_{\text{We drop this part, because all terms here is multiplied with x^4  and are > x^3 } }  \\ \text{we analyse } (1+x)^4(1+x^2)^4$$\\$\text{we expand }$ (1+\textcolor[rgb]{0,0,1}{4x} +6x^2+ \textcolor[rgb]{1,0,0}{4x^3} +x^4) ( \textcolor[rgb]{1,0,0}{1} +\textcolor[rgb]{0,0,1}{4x^2} +6x^4+4x^6+x^8 ) \\$\text{the terms with }$x^3$are$\textcolor[rgb]{0,0,1}{4x}*\textcolor[rgb]{0,0,1}{4x^2} $and$\textcolor[rgb]{1,0,0}{4x^3}*\textcolor[rgb]{1,0,0}{1} \\$the sum is$\textcolor[rgb]{0,0,1}{16x^3} + \textcolor[rgb]{1,0,0}{4x^3} = 20x^3\\$the coefficient of$x^3$is$20$\end{array} }}$$heureka Dec 5, 2014 #2 +27229 +5 Use the following to help: . Alan Dec 4, 2014 #3 +94114 +5 I did that too Alan but I am wondering how to get it by hand. Given time I shall work it out unless someone else works it out and shows me beforehand. Melody Dec 4, 2014 #4 +20680 +10 Best Answer What is the coefficient of in$$\\ \small{\text{ \begin{array}{l} \text{set }$ 1+x+x^2+x^3 = (1+x)(1+x^2) $\text{ we have }$\left[(1+x)(1+x^2)+x^4 \right]^4 $\\ \text{now we expand}$\\[(1+x)(1+x^2)]^4
\underbrace{
+ 4*[(1+x)(1+x^2)]^3(\textcolor[rgb]{1,0,0}{x^4})+6*[(1+x)(1+x^2)]^2(\textcolor[rgb]{1,0,0}{x^4})^2+4*[(1+x)(1+x^2)](\textcolor[rgb]{1,0,0}{x^4})^3+ (\textcolor[rgb]{1,0,0}{x^4})^4
}_{\text{We drop this part, because all terms here is multiplied with $x^4$ and are $> x^3$ } }
$\\ \text{we analyse }$ (1+x)^4(1+x^2)^4 $$\\\text{we expand }  (1+{4x} +6x^2+ {4x^3} +x^4) ( {1} +{4x^2} +6x^4+4x^6+x^8 )  \\\text{the terms with } x^3 are {4x}*{4x^2}  and {4x^3}*{1}  \\the sum is {16x^3} + {4x^3} = 20x^3 \\the coefficient of x^3 is 20 \end{array} }}$$

heureka  Dec 5, 2014
#5
+94114
0

Thanks Heureka

Melody  Dec 5, 2014