a) \({\left( {3x + y} \right)^4} = {\left( {3x} \right)^4} + 4.{\left( {3x} \right)^3}y + 6.{\left( {3x} \right)^2}{y^2} + 4.\left( {3x} \right){y^3} + {y^4}\)

\( = 81{x^4} + 108{x^3}y + 54{x^2}{y^2} + 12x{y^3} + {y^4}\)

b) \(\begin{array}{l}{\left( {x - \sqrt 2 } \right)^5} = \left( {x + (-\sqrt 2) } \right)^5 ={x^5} + 5.{x^4}.\left( { - \sqrt 2 } \right) + 10.{x^3}.{\left( { - \sqrt 2 } \right)^2} + 10.{x^2}.{\left( { - \sqrt 2 } \right)^3} + 5.x.{\left( { - \sqrt 2 } \right)^4} + 1.{\left( { - \sqrt 2 } \right)^5}\\ = {x^5} - 5\sqrt 2 .{x^4} + 20{x^3} - 20\sqrt 2 .{x^2} + 20x - 4\sqrt 2 \end{array}\)

(x^2+1/x)^4

\(=C^0_4\cdot\left(x^2\right)^4+C^1_4\cdot\left(x^2\right)^3\cdot\left(\dfrac{1}{x}\right)+C^2_4\cdot\left(x^2\right)^2\cdot\left(\dfrac{1}{x}\right)^2+C^3_4\cdot\left(x^2\right)^1\cdot\left(\dfrac{1}{x}\right)^3+C^4_4\cdot\left(x^2\right)^0\cdot\left(\dfrac{1}{x}\right)^4\)

=x^8+4x^5+6x^3+4/x+1/x^4

có anh ạ

SHTQ của \(\left(3x+2\right)^5\) là \(C^k_5\cdot\left(3x\right)^{5-k}\cdot2^k=C^k_5\cdot3^{5-k}\cdot2^k\cdot x^{5-k}\)

Hệ số của số hạng chứa x tương ứng với 5-k=1

=>k=4

=>Hệ số là \(C^4_5\cdot3^{5-4}\cdot2^4=240\)

Xét khai triển:

\(\left(1+x\right)^n=C_n^0+C_n^1x+C_n^2x^2+...+C_n^nx^n\)

\(\Leftrightarrow x\left(1+x\right)^n=C_n^0x+C_n^1x^2+C_n^2x^3+...+C_n^nx^{n+1}\)

Đạo hàm 2 vế:

\(\left(1+x\right)^n+nx\left(1+x\right)^{n-1}=C_n^0+2C_n^1x+3C_n^2x^2+...+\left(n+1\right)C_n^nx^n\)

Thay \(x=1\)

\(\Rightarrow2^n+n.2^{n-1}=1+2C_n^1+3C_n^2+...+\left(n+1\right)C_n^n\)

\(\Rightarrow2^{n-1}\left(2+n\right)-1=111\)

\(\Rightarrow2^{n-1}\left(2+n\right)=112=2^4.7\)

\(\Rightarrow n=5\)

\(\left(x^2+\dfrac{2}{x}\right)^5=\sum\limits^5_{k=0}C_5^kx^{2k}.2^{5-k}.x^{k-5}=\sum\limits^5_{k=0}C_5^k.2^{5-k}.x^{3k-5}\)

\(3k-5=4\Rightarrow k=3\Rightarrow\) hệ số: \(C_5^3.2^2\)

\(\left(2x+1\right)\left(x+3\right)+\left(x+1\right)^2\left(x+2\right)+\left(x+5\right)\left(x+1\right)\)

\(=2x^2+6x+x+3+x^3+2x^2+x+2x^2+4x+2+x^2+x+5x+5\)

\(=x^3+7x^2+18x+10\)

đúng ko nhỉ?

tham khảo : KHAI TRIỂN RÚT GỌN ĐA THỨC BẰNG CASIO (1LINK DUY NHẤT) - YouTube

Đáp án là C