if a + b + c is equals to zero find the value of a cube plus b cube plus c cube
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Answered by
3
The answer is 3abc, Here is the proof :
According to the question, (a+b+c) = 0,
Now, Cubing on both the sides,
=> (a+b+c)³ = 0³,
=> a³ + 3a²(b+c) + 3a(b+c)² + (b+c)³,
=> a³ + 3a²(b+c) + 3a(b²+2bc+c²) + b³ + 3b²c + 3bc² + c³,
=>a³ + 3a²b + 3a²c + 3ab² + 6abc + 3ac² + b³ + 3b²c + 3bc² + c³,
Bringing all the likely terms together,
=> a³ + 3a²b + 3a²c + b³ + 3b²c + 3b²a + c³ + 3c²a + 3c²b + 6abc,
=> a³ + 3a²(b+c) + 3b²(c+a) + 3c²(a+b) + 6abc,
We know that a+b+c = 0 ,
=> a + b = -c , Similarly,
=> a + c = -b,
=> b + c = -a,
Substituting these values in the last simplified equation,
=> a³ + 3a²(-a) + b³ + 3b²(-b) + c³ + 3c²(-c) + 6abc,
=> a³ - 3a³ + b³ - 3b³ + c³ - 3c³ + 6abc ,
We know that, We got this equation by simplyfying (a+b+c)³ = 0³,
Which means, We can equal the last equation with 0,
=> -2a³ -2b³ -2c³ + 6abc = 0,
=> -2a³ -2b³ -2c³ = -6abc,
=> 2a³ + 2b³ + 2c³ = 6abc,
=> a³ + b³ + c³ = 3abc,
Therefore a³ + b³ + c³ = 3abc, Which means the answer is 3abc,
Hope you can understand, Have a Great day !,
Thanking you, Bunti 360 !
According to the question, (a+b+c) = 0,
Now, Cubing on both the sides,
=> (a+b+c)³ = 0³,
=> a³ + 3a²(b+c) + 3a(b+c)² + (b+c)³,
=> a³ + 3a²(b+c) + 3a(b²+2bc+c²) + b³ + 3b²c + 3bc² + c³,
=>a³ + 3a²b + 3a²c + 3ab² + 6abc + 3ac² + b³ + 3b²c + 3bc² + c³,
Bringing all the likely terms together,
=> a³ + 3a²b + 3a²c + b³ + 3b²c + 3b²a + c³ + 3c²a + 3c²b + 6abc,
=> a³ + 3a²(b+c) + 3b²(c+a) + 3c²(a+b) + 6abc,
We know that a+b+c = 0 ,
=> a + b = -c , Similarly,
=> a + c = -b,
=> b + c = -a,
Substituting these values in the last simplified equation,
=> a³ + 3a²(-a) + b³ + 3b²(-b) + c³ + 3c²(-c) + 6abc,
=> a³ - 3a³ + b³ - 3b³ + c³ - 3c³ + 6abc ,
We know that, We got this equation by simplyfying (a+b+c)³ = 0³,
Which means, We can equal the last equation with 0,
=> -2a³ -2b³ -2c³ + 6abc = 0,
=> -2a³ -2b³ -2c³ = -6abc,
=> 2a³ + 2b³ + 2c³ = 6abc,
=> a³ + b³ + c³ = 3abc,
Therefore a³ + b³ + c³ = 3abc, Which means the answer is 3abc,
Hope you can understand, Have a Great day !,
Thanking you, Bunti 360 !
Answered by
2
The answer is 3abc, Here is the proof :
According to the question, (a+b+c) = 0,
Now, Cubing on both the sides,
=> (a+b+c)³ = 0³,
=> a³ + 3a²(b+c) + 3a(b+c)² + (b+c)³,
=> a³ + 3a²(b+c) + 3a(b²+2bc+c²) + b³ + 3b²c + 3bc² + c³,
=>a³ + 3a²b + 3a²c + 3ab² + 6abc + 3ac² + b³ + 3b²c + 3bc² + c³,
Bringing all the likely terms together,
=> a³ + 3a²b + 3a²c + b³ + 3b²c + 3b²a + c³ + 3c²a + 3c²b + 6abc,
=> a³ + 3a²(b+c) + 3b²(c+a) + 3c²(a+b) + 6abc,
We know that a+b+c = 0 ,
=> a + b = -c , Similarly,
=> a + c = -b,
=> b + c = -a,
Substituting these values in the last simplified equation,
=> a³ + 3a²(-a) + b³ + 3b²(-b) + c³ + 3c²(-c) + 6abc,
=> a³ - 3a³ + b³ - 3b³ + c³ - 3c³ + 6abc ,
We know that, We got this equation by simplyfying (a+b+c)³ = 0³,
Which means, We can equal the last equation with 0,
=> -2a³ -2b³ -2c³ + 6abc = 0,
=> -2a³ -2b³ -2c³ = -6abc,
=> 2a³ + 2b³ + 2c³ = 6abc,
=> a³ + b³ + c³ = 3abc,
Therefore a³ + b³ + c³ = 3abc, Which means the answer is 3abc,
According to the question, (a+b+c) = 0,
Now, Cubing on both the sides,
=> (a+b+c)³ = 0³,
=> a³ + 3a²(b+c) + 3a(b+c)² + (b+c)³,
=> a³ + 3a²(b+c) + 3a(b²+2bc+c²) + b³ + 3b²c + 3bc² + c³,
=>a³ + 3a²b + 3a²c + 3ab² + 6abc + 3ac² + b³ + 3b²c + 3bc² + c³,
Bringing all the likely terms together,
=> a³ + 3a²b + 3a²c + b³ + 3b²c + 3b²a + c³ + 3c²a + 3c²b + 6abc,
=> a³ + 3a²(b+c) + 3b²(c+a) + 3c²(a+b) + 6abc,
We know that a+b+c = 0 ,
=> a + b = -c , Similarly,
=> a + c = -b,
=> b + c = -a,
Substituting these values in the last simplified equation,
=> a³ + 3a²(-a) + b³ + 3b²(-b) + c³ + 3c²(-c) + 6abc,
=> a³ - 3a³ + b³ - 3b³ + c³ - 3c³ + 6abc ,
We know that, We got this equation by simplyfying (a+b+c)³ = 0³,
Which means, We can equal the last equation with 0,
=> -2a³ -2b³ -2c³ + 6abc = 0,
=> -2a³ -2b³ -2c³ = -6abc,
=> 2a³ + 2b³ + 2c³ = 6abc,
=> a³ + b³ + c³ = 3abc,
Therefore a³ + b³ + c³ = 3abc, Which means the answer is 3abc,
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