How do you expand the binomial #(2x-y^3)^7# using the binomial theorem?

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Answer 1 · 2018-03-30T18:07:34

#128x^7-448x^6y^3+672x^5y^6-560x^4y^9+280x^3y^12-84x^2y^15#
#+14xy^18-y^21#

For the expansion of #(x+y)^n# we have:

#sum_(r=0)^n((n),(r))x^(n-r)y^r#

Where:

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

#(2x-y^3)^7#

#((7),(0))(2x)^7(-y^3)^0+((7),(1))(2x)^6(-y^3)^1+((7),(2))(2x)^5(-y^3)^2#

#+((7),(3))(2x)^4(-y^3)^3+((7),(4))(2x)^3(-y^3)^4+((7),(5))(2x)^2(-y^3)^5#

#+((7),(6))(2x)^1(-y^3)^6+((7),(7))(2x)^0(-y^3)^7#

Using # \ \ \ (n!)/((r!(n-r)!)#

#128x^7-448x^6y^3+672x^5y^6-560x^4y^9+280x^3y^12-84x^2y^15#
#+14xy^18-y^21#