Question

Chứng minh đẳng thức:

a, ( x-y . ( x+y ) = x² - y²

b, ( x-y ) . (x³+xy²+x²y+y³  )= x⁴-y⁴

c,(a+b+c). (ab+bc+ac)-abc=(a+b). (a+c) . (b+c)

Answer 1

a) \(\left(x-y\right)\left(x+y\right)\)

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

\(=x^2-y^2\)

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

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

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

c)\(\left(a+b+c\right)\left(ab+bc+ac\right)-abc\)

\(=a^2b+abc+a^2c+ab^2+b^2c+abc+abc+bc^2+ac^2-abc\)

\(=2abc+a^2b+a^2c+ab^2+b^2c+bc^2+ac^2\left(1\right)\)

\(\left(a+b\right)\left(a+c\right)\left(b+c\right)\)

\(=a^2+ac+ab+bc\left(b+c\right)\)

\(=a^2b+abc+ab^2+b^2c+a^2c+ac^2+abc+bc^2\)

\(=2abc+a^2b+ab^2+b^2c+a^2c+ac^2+bc^2\left(2\right)\)

Từ (1)(2) => đpcm

Answer 2

đẽ thu gọn vế vd a) ta có vt: ( x-y) .(x+y)=x^2 -y^2

                                                                 =vp

                                                               ->dpcm

b) (x-y) . (x^3 +xy^2 +x^2y+y^3)

  =(x-y ).(x^3 + y^3) 

= x.x^3 -y.y^3

=x^4 - y^4 =vp

->dpcm

c) (a +b+ c) (ab +bc +ac) -abc 

=nhân vô rút gọn 

=(a^2b +2abc +c^b) +(a^2c+c^2a) + (ab^2+b^2c )

=b(a+c)^2 +ac(a+c) +b^2 (a+c) 

=(a+c).[b(a+c)+b^2 +ac+b^2]

=(a+c)(ab+b^2+bc+ac)

=(a+c) [b(a+b)+c(a+b)]

=(a+b)(a+c)(b+c)=vp 

->dpcm