1.Sử dụng công thức nhị thức Newton,khai triển:$(6x-2y^3+y^2z+2w)^5$

Đáp án:

$7776x^5-12960x^4y^3+8640x^3y^6-2880x^2y^9+480xy^{12}-32y^{15}+6840x^4y^2z-8640x^3y^5z+4320x^2y^8z-960y^{11}z+80y^{14}z+12960x^4w-17280x^3y^3w+8640x^2y^6w-1920y^9w+160y^{12}w+2160x^3y^4z^2-2160x^2y^7z^2+720xy{10}z^2-80y^{13}z^2+8640x^3y^2zw-8640x^2y^5zw+2880xy^8zw-320y^{11}zw+8640x^3w^2-8640x^2y^3w^2+2880xy^6w^2-320y^9w^2+360x^2y^6z^3+2160x^2y^4z^2w+4320x^2y^2zw^2+2880x^2w^3-240xy^9z^3-1440xy^7z^2w-2880xy^5zw^2-1920xy^3w^3+40y^{12}z^3+240y^{10}z^2w+480y^8zw^2+320y^6w^3+30xy^8z^4+240xy^6z^3w+720xy^4z^2w^2+960xy^2zw^3+480xw4-10y^{11}z^4-80y^9z^3w-240y^7z^2w^2-320y^5zw^3-160y^3w^4+y^{10}z^5+10y^8z^4w+40y^6z^3w^2+80y^4z^2w^3+80y^2zw^4+32w^5$

Giải thích các bước giải:

Sử dụng công thức nhị thức Newton ,ta có:

  $(6x-2y^3+y^2z+2w)^5$

=$(6-2y^3)^5+5.(6x-2y^3)^4.(y^2z+2w)+10.(6x-2y^3)^3.(y^2z+2w)^2+10.(6x-2y^3)^2.(y^2z+2w)^3+5.(6x-2y^3).(y^2z+2w)^4+(y^2z+2w)^5$(*)

•$(6x-2y^3)^5=7776x^5-12960x^4y^3+8640x^3y^6-2880x^2y^9+480xy^{12}-32y^{15}$ (1)

•$(6x-2y^3)^4=1296x^4-1728x^3y^3+864x^2y^6-192y^9+16y^{12}$ (2)

•$(6x-2y^3)^3=216x^3-216x^2y^3+72xy^6-8y^9$ (3)

•$(y^2z+2w)^2=y^4z^2+4y^2zw+4w^2$ (4)

•$(6x-2y^3)^2=36x^2-24xy^3+4y^6$ (5)

•$(y^2z+2w)^3=y^6z^3+6y^4z^2w+12y^2zw^2+8w^3$ (6)

•$(y^2z+2w)^4=y^8z^4+8y^6z^3w+24y^4z^2w^2+32y^2zw^3+16w^4$ (7)

•$(y^2z+2w)^5=y^{10}z^5+10y^8z^4w+40y^6z^3w^2+80y^4z^2w^3+80y^2zw^4+32w^5$ (8)

Từ (1,2,3,4,5,6,7,8) ta thay vào (*),ta có:

$(6x-2y^3+y^2z+2w)^5$

=$7776x^5-12960x^4y^3+8640x^3y^6-2880x^2y^9+480xy^{12}-32y^{15}+6840x^4y^2z-8640x^3y^5z+4320x^2y^8z-960y^{11}z+80y^{14}z+12960x^4w-17280x^3y^3w+8640x^2y^6w-1920y^9w+160y^{12}w+2160x^3y^4z^2-2160x^2y^7z^2+720xy{10}z^2-80y^{13}z^2+8640x^3y^2zw-8640x^2y^5zw+2880xy^8zw-320y^{11}zw+8640x^3w^2-8640x^2y^3w^2+2880xy^6w^2-320y^9w^2+360x^2y^6z^3+2160x^2y^4z^2w+4320x^2y^2zw^2+2880x^2w^3-240xy^9z^3-1440xy^7z^2w-2880xy^5zw^2-1920xy^3w^3+40y^{12}z^3+240y^{10}z^2w+480y^8zw^2+320y^6w^3+30xy^8z^4+240xy^6z^3w+720xy^4z^2w^2+960xy^2zw^3+480xw4-10y^{11}z^4-80y^9z^3w-240y^7z^2w^2-320y^5zw^3-160y^3w^4+y^{10}z^5+10y^8z^4w+40y^6z^3w^2+80y^4z^2w^3+80y^2zw^4+32w^5$