What is the coefficient of x^2 of (x-2)^7?

3 Answers
Mar 1, 2018

#-672#

Explanation:

#(x-2)^7=x^7 - 7*2^1 x^6 + 21*2^2 x^5 - 35*2^3 x^4 + 35*2^4 x^3 - 21*2^5 x^2 + 7*2^6 x - 2^7#
#- 21*2^5 x^2=-672x^2#

#"The coefficient of " x^2" in " (x-2)^7" is " -672#

Explanation:

#(x-2)^7=^7C_0x^7(-2)^0+^7C_1x^6(-2)^1+^7C_2x^5(-2)^2+^7C_3x^4(-2)^3+^7C_4x^3(-2)^4+^7C_5x^2(-2)^5+^7C_6x^1(-2)^6+^7C_7x^0(-2)^7#

#^7C_0=1#
#^7C_1=7#
#^7C_2=21#
#^7C_3=35#
#^7C_4=35#
#^7C_5=21#
#^7C_6=7#
#^7C_7=1#

#(-2)^0=1#
#(-2)^1=-2#
#(-2)^2=4#
#(-2)^3=-8#
#(-2)^4=16#
#(-2)^5=-32#
#(-2)^6=64#
#(-2)^7=-128#

#(x-2)^7=1x^7(1)+7x^6(-2)+21x^5(4)+35x^4(-8)+35x^3(16)+21x^2(-32)+7x^1(64)+^1x^0(-128)#

#(x-2)^7#
#=x^7-14x^6+84x^5-280x^4-560x^3-672x^2+448x-128#

#"The coefficient of " x^2" in " (x-2)^7" is " -672#

Mar 1, 2018

#-672#

Explanation:

For binomial expansion of #(a +b)^n#:

Each term is:

#((n),(r))a^(n-r)b^(r)#

Where:

#((n),(r))=color(white)(0)^nC_(r)=(n!)/(r!(n-r)!#

So the coefficient of #bb(x^2)# is:

For #x^2# we have:

#7-r=2=>r=5#

#((7),(5))(x)^(7-5)(-2)^(5)#

#((7),(5))(x)^2(-2)^(5)=(21)(x^2)(-32)=-672#