Giải bài tập 1 trang 37 Chuyên đề toán 10 cánh diều

a) $(2x+y)^{6}$

$=C_{6}^{0}(2x)^{6}+C_{6}^{1}(2x)^{5}y+C_{6}^{2}(2x)^{4}y^{2}+C_{6}^{3}(2x)^{3}y^{3}+C_{6}^{4}(2x)^{2}y^{4}+C_{6}^{5}(2x)y^{5}+C_{6}^{6}y^{6}$

$=2^{6}x^{6}+C_{6}^{1}2^{5}x^{5}y+C_{6}^{2}2^{4}x^{4}y^{2}+C_{6}^{3}2^{3}x^{3}y^{3}+C_{6}^{4}2^{2}x^{2}y^{4}+C_{6}^{5}2xy^{5}+C_{6}^{6}y^{6}$

b) $(x-3y)^{6}=[x+(-3y)]^{6}$

$=C_{6}^{0}x^{6}+C_{6}^{1}x^{5}(-3y)+C_{6}^{2}x^{4}(-3y)^{2}+C_{6}^{3}x^{3}(-3y)^{3}+C_{6}^{4}x^{2}(-3y)^{4}+C_{6}^{5}x(-3y)^{5}+C_{6}^{6}(-3y)^{6}$

$=x^{6}-C_{6}^{1}3x^{5}y+C_{6}^{2}3^{2}x^{4}y^{2}-C_{6}^{3}3^{3}x^{3}y^{3}+C_{6}^{4}3^{4}x^{2}y^{4}+C_{6}^{5}3^{5}xy^{5}+3^{6}y^{6}$

c) $(x-1)^{n}=[x+(-1)]^{n}$

$=C_{n}^{0}x^{n}+C_{n}^{1}x^{n-1}(-1)+C_{n}^{2}x^{n-2}(-1)^{2}+...+C_{n}^{n-1}x(-1)^{n-1}+C_{n}^{n}(-1)^{n}$

$=x^{n}+C_{n}^{1}(-1)x^{n-1}+C_{n}^{2}(-1)^{2}x^{n-2}+...+C_{n}^{n-1}(-1)^{n-1}x+(-1)^{n}$

d) $(x+2)^{n}$

$=C_{n}^{0}x^{n}+C_{n}^{1}x^{n-1}2+C_{n}^{2}x^{n-2}2^{2}+...+C_{n}^{n-1}x2^{n-1}+C_{n}^{n}2^{n}$

 $=x^{n}+C_{n}^{1}2x^{n-1}+C_{n}^{2}2^{2}x^{n-2}+...+C_{n}^{n-1}2^{n-1}+2^{n}$

e) $(x+y)^{2n}$

$=C_{2n}^{0}x^{2n}+C_{2n}^{1}x^{2n-1}y+C_{2n}^{2}x^{2n-2}y^{2}+...+C_{2n}^{2n-1}xy^{2n-1}+C_{2n}^{2n}y^{2n}$

$=x^{2n}+C_{2n}^{1}x^{2n-1}y+C_{2n}^{2}x^{2n-2}y^{2}+...+C_{2n}^{2n-1}xy^{2n-1}+y^{2n}$

g) $(x-y)^{2n}$

$=C_{2n}^{0}x^{2n}+C_{2n}^{1}x^{2n-1}(-y)+C_{2n}^{2}x^{2n-2}(-y)^{2}+...+C_{2n}^{2n-1}x(-y)^{2n-1}+C_{2n}^{2n}(-y)^{2n}$

$=x^{2n}-C_{2n}^{1}y+C_{2n}^{2}x^{2n-2}y^{2}-...-C_{2n}^{2n-1}xy^{2n-1}+y^{2n}$