How do you expand the binomial #(2x-y)^6# using the binomial theorem?

Answer 1

#(2x - y)^6 = 64x^6 - 192x^5y + 240x^4y^2 - 160x^3y^3 + 60x^2y^4 - 12xy^5 + y^6#

#(2x - y)^6#

The binomial theorem states that for any binomial #(a + b)^n#, the general expansion is given by #(a + b)^n = color(white)(two)_nC_r xx a^(n - r) xx b^r#, where #r# is in ascending powers from #0# to #n# and #n# is in descending powers from #n# to #0#.

# = color(white)(two)_6C_0(2x)^6(-y)^0 + color(white)(two)_6C_1(2x)^5(-y)^1 + color(white)(two)_6C_2(2x)^4(-y)^2 + color(white)(two)_6C_3(2x)^3(-y)^3 + color(white)(two)_6C_4(2x)^2(-y)^4 + color(white)(two)_6C_5(2x)^1(-y)^5 + color(white)(two)_6C_6(2x)^0(-y)^6 #

#=64x^6 - 192x^5y + 240x^4y^2 - 160x^3y^3 + 60x^2y^4 - 12xy^5 + y^6#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.