Question

prove the following (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx​

Answer 1

Step-by-step explanation:

Let x+y = A

So, (x+y+z)² becomes (A+z)²

But by identity , (a+b)² = a² + b² + 2ab

So,

(A+z)² = A² + z² + 2Az

Now substituting A = x + y in (A+z)² = A² + z² + 2Az

(x+y+z)² = (x+y)² + z² + 2*(x+y)*z

(x+y+z)² = (x+y)² + z² + 2xz + 2yz

But, (x+y)² = x² + y² + 2xy

So,

(x+y+z)² = x² + y² + 2xy + z² + 2xz + 2yz

Therefore ,

(x+y+z)² = x² + y² + z² + 2xy + 2xz +2zx

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Answer 2

Step-by-step explanation:

(x+y+z)² = (x+y+z)(x+y+z)

x(x+y+z)+y(x+y+z)+z(x+y+z)

x²+xy +xz +xy + y² +yz + xz +zy+ z²

= x²+y²+z²+2xy +2xz + 2zx l.h.s = r.h.s

hence proved.

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