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Find the x^4 term of (5x-3)^10

 Apr 6, 2018
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Find the x^4 term of (5x-3)^10

 

\(\begin{array}{|rcll|} \hline && (5x-3)^{10} \\ &=& \dbinom{10}{0}(5x)^{10} +\dbinom{10}{1}(5x)^{9}(-3)^1 +\dbinom{10}{2}(5x)^{8}(-3)^2 +\dbinom{10}{3}(5x)^{7}(-3)^3\\ &+& \dbinom{10}{4}(5x)^{6}(-3)^4 +\dbinom{10}{5}(5x)^{5}(-3)^5 +{\color{red}{\dbinom{10}{6}(5x)^{4}(-3)^6}} + \ldots +\dbinom{10}{10}(-3)^{10} \\ \hline \end{array} \)

 

 

\(\text{$x^4$ term:}\)

\(\begin{array}{|rcll|} \hline && \dbinom{10}{6}(5x)^{4}(-3)^6 \\\\ &=& \dbinom{10}{10-6}5^4x^43^6 \\\\ &=& \dbinom{10}{4}3^65^4x^4 \\\\ &=& \dfrac{10}{4}\cdot \dfrac{9}{3}\cdot \dfrac{8}{2}\cdot \dfrac{7}{1}\cdot 3^65^4x^4 \\\\ &=& 10\cdot 3 \cdot 7 \cdot 3^65^4x^4 \\\\ &=& 2 \cdot 5\cdot 3 \cdot 7 \cdot 3^65^4x^4 \\\\ &=& 2\cdot 7\cdot 3^75^5x^4 \\\\ &=& 14\cdot 3^75^5x^4 \\\\ &=& 14\cdot 2187\cdot 3125 x^4 \\\\ &=& 95681250 x^4 \\ \hline \end{array}\)

 

 

laugh

 Apr 6, 2018

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