Oral Quest

Algebra Level 3

A = ( w + x ) ( y − z ) + ( w + y ) ( z − x ) + ( w + z ) ( x − y ) B = ( w − x ) ( y − z ) + ( w − y ) ( z − x ) + ( w − z ) ( x − y ) \large{A=(w+x)(y-z)+(w+y)(z-x)+(w+z)(x-y) \\ B=(w-x)(y-z)+(w-y)(z-x)+(w-z)(x-y)}

Compare A , B A,B for w , x , y , z ∈ R w,x,y,z \in \mathbb{R}

A > B A>B A = B A=B Cannot be determined A < B A<B

Nihar Mahajan by Nihar Mahajan , 20, India

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2 solutions

Nihar Mahajan
Jul 29, 2015

A = ( w + x ) ( y − z ) + ( w + y ) ( z − x ) + ( w + z ) ( x − y ) = w y − w z + x y − x z + w z − w x + y z − x y + w x − w y − x z − z y ⇒ A = 0 B = ( w − x ) ( y − z ) + ( w − y ) ( z − x ) + ( w − z ) ( x − y ) = w y − w z − x y + x z + w z − w x − y z + x y + w x − w y − x z + z y ⇒ B = 0 A = B A=(w+x)(y-z)+(w+y)(z-x)+(w+z)(x-y)\\ = wy-wz+xy-xz+wz-wx+yz-xy+wx-wy-xz-zy \\ \Rightarrow A=0 \\ B=(w-x)(y-z)+(w-y)(z-x)+(w-z)(x-y) \\ = wy-wz-xy+xz+wz-wx-yz+xy+wx-wy-xz+zy \\ \Rightarrow B=0 \\ \large \boxed{A=B}

3 Helpful 0 Interesting 0 Brilliant 0 Confused
K T
Aug 20, 2020

When expanding, all terms vanish, so A = B = 0 A = B = 0 .

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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