State true or false.The expression

a−b

3

+

b−c

3

+

c−a

3

can be factored as

3(a−b)(b−c)(c−a)
can be factored as .

Question

State true or false.The expression

a−b
     
    3
   
   +

b−c
     
    3
   
   +

c−a
     
    3

can be factored as

3(a−b)(b−c)(c−a)
  can be factored as .

Accepted Answer

We have to check if the expression

a−b
     
    3
   
   +

b−c
     
    3
   
   +

c−a
     
    3

can be factored as

3(a−b)(b−c)(c−a)
 .Using the formulas,

(a+b)
    2
   
   =
    a
    2
   
   +
    b
    2
   
   +2ab
 ,

(a−b)
    2
   
   =
    a
    2
   
   +
    b
    2
   
   −2ab
 ,

a
    2
   
   −
    b
    2
   
   =(a+b)(a−b)
 ,

a
    3
   
   −
    b
    3
   
   =(a−b)(
    a
    2
   
   +
    b
    2
   
   +ab)
 ,

a
    3
   
   +
    b
    3
   
   +
    c
    3
   
   −3abc=(a+b+c)(
    a
    2
   
   +
    b
    2
   
   +
    c
    2
   
   −ab−bc−ca)
 Computing by using the formula of

a
    3
   
   +
    b
    3
   
   +
    c
    3
   
   −3abc
 ,

a
    3
   
   +
    b
    3
   
   +
    c
    3
   
   −3abc=(a+b+c)(
    a
    2
   
   +
    b
    2
   
   +
    c
    2
   
   −ab−bc−ca)

when,(a+b+c)=0

a
    3
   
   +
    b
    3
   
   +
    c
    3
   
   =3abc
 ..................(1) Assuming,

X=(a−b),Y=(b−c),
  and

Z=(c−a),
  Then,

X+Y+Z=a−b+b−c+c−a

=0
  Using equation (1),

(a−b)
    3
   
   +
    
     (b−c)
    3
   
   +
    
     (c−a)
    3
   
   =3(a−b)(b−c)(c−a)

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

=
    a
    2
   
   c−
    a
    2
   
   b+a
    b
    2
   
   −a
    c
    2

−3(a−b)(b−c)(c−a)=−3(a−b)(bc−ab−
    c
    2
   
   +ac)

=−3(
    a
    2
   
   c−
    a
    2
   
   b+a
    b
    2
   
   −a
    c
    2
   
   )

a
    3
   
   +
    b
    3
   
   +
    c
    3
   
   −3abc=(a+b+c)(
    a
    2
   
   +
    b
    2
   
   +
    c
    2
   
   −ab−bc−ca)
 ..........(2) From equation (2),

a+b+c

a
      2
     
     +
      b
      2
     
     +
      c
      2
     
     −ab−bc−ca
   =
    a
    3
   
   +
    b
    3
   
   +
    c
    3
   
   −3abc
 Thus, the given statement is true.Therefore, option 1 is correct.

State true or false.

The expression $a − b 3 + b − c 3 + c − a 3$ can be factored as $3 (a − b) (b − c) (c − a)$ can be factored as .

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State true or false.The expression a−b 3 + b−c 3 + c−a 3 can be factored as 3(a−b)(b−c)(c−a) can be factored as .