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What is the coefficient of x^3 in (x^4 + x^3 + x^2 + x + 1)^4

Guest Dec 4, 2014

Best Answer 

 #4
avatar+18845 
+10

What is the coefficient of x^3 in (x^4 + x^3 + x^2 + x + 1)^4

 $$\\ \small{\text{
\begin{array}{l}
\text{set } $ 1+x+x^2+x^3 = (1+x)(1+x^2) $
\text{ we have }$\left[(1+x)(1+x^2)+x^4 \right]^4 $\\
\text{now we expand} $\\$ $[(1+x)(1+x^2)]^4
\underbrace{
+ 4*[(1+x)(1+x^2)]^3(\textcolor[rgb]{1,0,0}{x^4})+6*[(1+x)(1+x^2)]^2(\textcolor[rgb]{1,0,0}{x^4})^2+4*[(1+x)(1+x^2)](\textcolor[rgb]{1,0,0}{x^4})^3+ (\textcolor[rgb]{1,0,0}{x^4})^4
}_{\text{We drop this part, because all terms here is multiplied with $x^4 $ and are $> x^3$ } }
$ \\
\text{we analyse }$ (1+x)^4(1+x^2)^4 $$\\$\text{we expand } $ (1+\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
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4+0 Answers

 #2
avatar+26412 
+5

 

Use the following to help:

 binomial expansion

.

Alan  Dec 4, 2014
 #3
avatar+91517 
+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
avatar+18845 
+10
Best Answer

What is the coefficient of x^3 in (x^4 + x^3 + x^2 + x + 1)^4

 $$\\ \small{\text{
\begin{array}{l}
\text{set } $ 1+x+x^2+x^3 = (1+x)(1+x^2) $
\text{ we have }$\left[(1+x)(1+x^2)+x^4 \right]^4 $\\
\text{now we expand} $\\$ $[(1+x)(1+x^2)]^4
\underbrace{
+ 4*[(1+x)(1+x^2)]^3(\textcolor[rgb]{1,0,0}{x^4})+6*[(1+x)(1+x^2)]^2(\textcolor[rgb]{1,0,0}{x^4})^2+4*[(1+x)(1+x^2)](\textcolor[rgb]{1,0,0}{x^4})^3+ (\textcolor[rgb]{1,0,0}{x^4})^4
}_{\text{We drop this part, because all terms here is multiplied with $x^4 $ and are $> x^3$ } }
$ \\
\text{we analyse }$ (1+x)^4(1+x^2)^4 $$\\$\text{we expand } $ (1+\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
 #5
avatar+91517 
0

Thanks Heureka     

Melody  Dec 5, 2014

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