The coefficient of x^2 in the expansion of th | vedclass

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(A) Let $f(x) = (1 + 2x + 3x^2)^6 + (1 - 4x^2)^6$.We need the coefficient of $x^2$ in $(2 - x^2)f(x)$.This is equal to $2 	imes (	ext{coefficient of

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$106$

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(A) Let $f(x) = (1 + 2x + 3x^2)^6 + (1 - 4x^2)^6$.We need the coefficient of $x^2$ in $(2 - x^2)f(x)$.This is equal to $2 	imes (	ext{coefficient of } x^2 	ext{ in } f(x)) - 1 	imes (	ext{constant term in } f(x))$.First,find the constant term in $f(x)$:Constant term $= (1 + 2(0) + 3(0)^2)^6 + (1 - 4(0)^2)^6 = 1^6 + 1^6 = 2$.Next,find the coefficient of $x^2$ in $f(x)$:For $(1 + 2x + 3x^2)^6$,using the multinomial expansion or binomial expansion $(1 + (2x + 3x^2))^6 = 1 + 6(2x + 3x^2) + \binom{6}{2}(2x)^2 + \dots = 1 + 12x + 18x^2 + 15(4x^2) + \dots = 1 + 12x + 78x^2 + \dots$The coefficient of $x^2$ is $78$.For $(1 - 4x^2)^6$,the expansion is $1 + 6(-4x^2) + \dots = 1 - 24x^2 + \dots$The coefficient of $x^2$ is $-24$.Thus,the coefficient of $x^2$ in $f(x)$ is $78 - 24 = 54$.Finally,the coefficient of $x^2$ in $(2 - x^2)f(x)$ is $2(54) - 1(2) = 108 - 2 = 106$.

The coefficient of $x^2$ in the expansion of the product $(2 - x^2)((1 + 2x + 3x^2)^6 + (1 - 4x^2)^6)$ is

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