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What is the coeffiicient of x^2 in (3 + 5x)^7?

 Dec 6, 2019
 #1
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+1

Expand  (3 + 5x)^7

 

78125 x^7 + 328125 x^6 + 590625 x^5 + 590625 x^4 + 354375 x^3 + 127575 x^2 + 25515 x + 2187
(8 terms)

 Dec 6, 2019
 #2
avatar+7763 
+1

\(\quad (3 + 5x)^7\\ = \displaystyle \sum^7_{k = 0}\binom7k \cdot 3^{7 - k} \cdot (5x)^k\\ = \displaystyle \sum^7_{k = 0} \binom7k \cdot 3^7 \cdot \left(\dfrac{5}{3}\right)^k\cdot x^k\\ \text{When }k = 2,\\ \quad\text{Term}\\ = \displaystyle\binom72 \cdot 3^7 \cdot \left(\dfrac{5}{3}\right)^2 \cdot x^2\\ = 21 \cdot 243 \cdot 25 x^2\\ \quad\text{Coefficient of }x^2\\ = 21\cdot 243\cdot 25\\ =127575\)

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 Dec 6, 2019
 #3
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Max: How do you get :(5/3)^2 =25? . It should be just 5^2 =25.

 Dec 6, 2019
 #4
avatar+7763 
+1

I skipped a step. I will show it here.

 

\(\cdots\\ =\displaystyle \binom72 \cdot 3^7 \cdot \left(\dfrac{5}{3}\right)^2 \cdot x^2\\ \boxed{=\displaystyle \color{red}\binom72\color{black}\cdot \color{blue}3^5 \color{black}\cdot \color{green}5^2\color{black} \cdot x^2}\\ =\color{red} 21 \color{black}\cdot \color{blue}243\color{black}\cdot \color{green}25\color{black}\cdot x^2\)

MaxWong  Dec 6, 2019

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