- #1
superduck
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Homework Statement
I'm learning mathematics in Japanese, so maybe my English is not correct to discribe what I want to mention.
I'm struggling a development probloem below
( x + y + 2z)3 – ( y + 2z – x)3 – ( 2z + x – y )3 – ( x + y – 2z )3
Homework Equations
I used cube development formula like a3 + b3= ( a+ b )3 -3ab( a + b)
The Attempt at a Solution
Homework Statement
I solved this problem, but the answer of my reference book used another way to solve it. Then, I can't understand that way.
My attempt at a solution is below
transpose   {x + y + 2z}3 - { y+2z - x }3=P,
{ 2z + x - y }3 + { x + y - 2z}3=Q
P=[{( y+2z) + x } - {( y + 2z) - x}]3 + 3{( y + 2z ) + x}{( y + 2Z) - x}{( y + 2z + x) - ( y + 2z - x)}
=(2x)3 + 3{( y + 2z )2 - x2} ・2x
=8x3 + 6x( y2 + 4yz + 4z2 - x2)
=8x3 - 6x3 + 6xy2 + 24xyz + 24z2x
=2x3 + 6xy2 + 24z2x - 24xyzQ=[{ x -( y - 2z)} + { x + ( y - 2z)}]3 - 3{ x - ( y - 2z)}{ x + ( y - 2z)}×( x - y + 2z + x + y -2z)
=(2x)3 - 3{ x2 - ( y - 2z)2}・2x
=8x3 - 6x( x2 - y2 + 4yz - 4z2)
=2x3 - 6xy2 + 24z2x - 24xyz
thus,
P-Q=( 2x3 + 6xy2 + 24z2x + 24xyz) - ( 2x3 + 6xy2 + 24z2x - 24xyz)
   =48xyz
Homework Equations
I solved this problem like above, but the answer of the reference book is different from mine.
I can't this way to solve this problem. Please help me.
The answer of reference book is below
transpose   {x + y + 2z}3 - { y+2z - x }3=P,
{ 2z + x - y }3 + { x + y - 2z}3=Q
P={( x + y + 2z) - ( y + 2z - x)}{( x + y + 2z)2 + ( x + y +2z )( y + 2z - x) + ( y + 2z - x)2}
=2x{ x2 + 2x( y + 2z) + ( y + 2z)2 + ( y + 2z)2 - x2 + (y + 2z)2 - 2x( y + 2z) + x2}
=2x{3( y + 2z)2 + x2}
Q={(2z + x - y) + ( x + y - 2z)}{(2z + x - y)2 - ( 2z + x - y)( x + y - 2z) + (x + y - 2z)2}
=2x{(2z - y)2 + 2x(2z - y) + x2 - x2 + (2z - y)2 + (2z - y)2 - 2x( 2z - y) + x2}
=2x{3(2z - y)2 + x2}
thus,
P-Q=2x{3(y + 2z)2 + x2 - 3( 2z - y)2- x2}
=6x・8yz=48xyz
the answer is same to my result, but the way to solve this problem is different. I want to understand the solving way of reference book. Please help me.
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