+0

# What is the coefficient of in

0
198
4

What is the coefficient of  in

Guest Dec 4, 2014

#4
+18829
+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} }}$$heureka Dec 5, 2014 Sort: ### 4+0 Answers #2 +26402 +5 Use the following to help: . Alan Dec 4, 2014 #3 +91454 +5 I did that too Alan but I am wondering how to get it by hand. Given time I shall work it out unless someone else works it out and shows me beforehand. Melody Dec 4, 2014 #4 +18829 +10 Best Answer What is the coefficient of in$$\\ \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
+91454
0

Thanks Heureka

Melody  Dec 5, 2014

### 3 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details