State if the statement given below is true or false.The polynomial expression

25
a
2

−4
b
2

+28bc−49
c
2

, is equivalent to the product

5a−2b+7c

5a+2b−7c

.

Question

State if the statement given below is true or false.The polynomial expression

25
    a
    2
   
   −4
    b
    2
   
   +28bc−49
    c
    2

, is equivalent to the product

5a−2b+7c

5a+2b−7c
   
 .

Accepted Answer

A statement is given that the algebraic expression is

25−
    a
    2
   
   −
    b
    2
   
   −2ab
 , and is equivalent to

5a−2b+7c

5a+2b−7c
   
 . The algebraic expression can be regrouped as follows,Now, we know about the following identity,

a
    2
   
   +
    b
    2
   
   −2ab=

a−b
     
    2

. Using this, we can rewrite the expression as follows,

5a
     
    2
   
   −
    
     4
      b
      2
     
     +49
      c
      2
     
     −28bc
   ⇔

5
     
    2
   
   −

2b−7c
     
    2

It can be observed that the above expression is of the general form

x
      2
     
     −
      y
      2

, where

x=5a
  and

y=
    
     2b−7c
   
 .To factorize this expression, we can use the identity

a
    2
   
   −
    b
    2
   
   =
    
     a+b

a−b
   
 . So, using the identity, the expression

5a
     
    2
   
   −

2b−7c
     
    2

, can be written as follows,

5a
      
      2
     
     −

2b−7c
      
      2
     
     =
      
       5a+
        
         2b−7c

5a−
        
         2b−7c

=
      
       5a+2b−7c

5a−2b+7c

So, it can be observed that

25
    a
    2
   
   −4
    b
    2
   
   +28bc−49
    c
    2
   
   ⇔
    
     5a−2b+7c

5a+2b−7c
   
 .Hence, the given statement is true.

State if the statement given below is true or false.

The polynomial expression $25 a 2 − 4 b 2 + 28 b c − 49 c 2$ , is equivalent to the product $5 a − 2 b + 7 c 5 a + 2 b − 7 c$ .

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State if the statement given below is true or false.The polynomial expression 25 a 2 −4 b 2 +28bc−49 c 2 , is equivalent to the product 5a−2b+7c 5a+2b−7c .

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State if the statement given below is true or false.

The polynomial expression $25 a 2 − 4 b 2 + 28 b c − 49 c 2$ , is equivalent to the product $5 a − 2 b + 7 c 5 a + 2 b − 7 c$ .