Question

Cmr

a) \(\left(x-1\right)\left(x^2+x+1\right)=x^3-1\)

b)\(\left(x^3+x^2y+xy^2+y^3\right)\left(x-y\right)=x^4-y^4\)

c) \(\left(x+y+z\right)^2=x^2+y^2+z^2+2xy+2yz+2zx\)

d) \(\left(x+y+z\right)^3=x^3+y^3+z^3+3\left(x+y\right)\left(y+z\right)\left(z+x\right)\)

Answer 1

Câu a :

\(VT=\) \(\left(x-1\right)\left(x^2+x+1\right)=x^3-1^3=VP\)

Câu b :

\(VT=\)\(\left(x^3+x^2y+xy^2+y^3\right)\left(x-y\right)=x^4-y^4=VP\)

Tương tự bạn khai triển là ra nhé

Answer 2

a) \(\left(x-1\right)\left(x^2+x+1\right)\)

=\(x^3+x^2+x-x^2-x-1=x^3-1\)

\(\RightarrowĐPCM\)

b)\(\left(x^3+x^2y+xy^2+y^3\right)\left(x-y\right)\)

\(=x^4-x^3y+x^3y-x^2y^2+x^2y^2-xy^3+xy^3-y^4=x^4-y^4\)

Answer 3

c)(x+y+z)² = [(x + y) + z]² = (x + y)² + 2(x + y)z + z²

= x²+ 2xy + y² + 2xz + 2yz + z²

= x² + y² + z² + 2xy + 2yz + 2xz

Answer 4

a) \(\left(x-1\right)\left(x^2+x+1\right)=x^3-1^3\)

\(\Leftrightarrow x^3+x^2+x-x^2-x-1=x^3-1^3\)

\(\Leftrightarrow x^3-1=x^3-1^3\)

\(\Rightarrow\left(x-1\right)\left(x^2+x+1\right)=x^3-1^3\)

b) \(\left(x^3+x^2y+xy^2+y^3\right)\left(x-y\right)=x^4-y^4\)

\(\Leftrightarrow x^4+x^3y+x^2y^2+xy^3-x^3y-x^2y^2-xy^3-y^4=x^4-y^4\)

\(\Leftrightarrow x^4-y^4=x^4-y^4\)

\(\Rightarrow\left(x^3+x^2y+xy^2+y^3\right)\left(x-y\right)=x^4-y^4\)