What is the coefficient of #x^4# in the expansion of #(2+4x)^5(2-x)^4 #?

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Feb 13, 2018

Answer:

#-12768# is the coefficient of #x^4#

Explanation:

#(2+4x)^5(2-x)^4#
#((2+4x)^5)(2-x)^4=((5C0)2^5(4x)^0+(5C1)2^4(4x)^1+(5C2)2^3(4x)^2+(5C3)2^2(4x)^3+(5C4)2^1(4x)^4+(5C5)2^0(4x)^5)((4c0)2^4x^0-(4c1)2^3x^1+(4c2)2^2x^2-(4c3)2^1x^3+(4c4)2^0x^4)#
#((1)(32)(1)+(5)(16)(4x)+(10)(8)(16x^2)+(10)(4)(64x^3)+(5)(2)(256x^4)+(1)(1)(1024x^5))((1)(16)(x)^0-(4)(8)(x)^1+(6)(4)(x)^2-(4)(2)(x)^3+(1)(1)(x)^4)#
#(32+320x+1280x^2+2560x^3+2560x^4+1024x^5)(16-32x+24x^2-8x^3+x^4)#

Tabulating the products of coefficients
x 16 -32 24 -8 1
32 512 -1024 768 -256 32
320 5120 -10240 7680 -2560 320
1280 20480 -40960 30720 -10240 1280
2560 40960 -81920 61440 -20480 2560
2560 40960 -81920 61440 -20480 2560
1024 16384 -32768 24576 -8192 1024

The highlighted numbers are the coefficients of #x^4#
Adding them up,
#-12768# is the coefficient of #x^4#

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