Factor 343x^12 – 125y^27

 (7x - 3) • (49x2 + 21x + 9)

Step-by-step explanation:

STEP

1

:

Equation at the end of step 1

 73x3 -  27

STEP

2

:

Trying to factor as a Difference of Cubes:

2.1      Factoring:  343x3-27  

Theory : A difference of two perfect cubes,  a3 - b3 can be factored into

             (a-b) • (a2 +ab +b2)

Proof :  (a-b)•(a2+ab+b2) =

           a3+a2b+ab2-ba2-b2a-b3 =

           a3+(a2b-ba2)+(ab2-b2a)-b3 =

           a3+0+0-b3 =

           a3-b3

Check :  343  is the cube of  7  

Check :  27  is the cube of   3  

Check :  x3 is the cube of   x1

Factorization is :

            (7x - 3)  •  (49x2 + 21x + 9)  

Trying to factor by splitting the middle term

2.2     Factoring  49x2 + 21x + 9  

The first term is,  49x2  its coefficient is  49 .

The middle term is,  +21x  its coefficient is  21 .

The last term, "the constant", is  +9  

Step-1 : Multiply the coefficient of the first term by the constant   49 • 9 = 441  

Step-2 : Find two factors of  441  whose sum equals the coefficient of the middle term, which is   21 .

     -441    +    -1    =    -442  

     -147    +    -3    =    -150  

     -63    +    -7    =    -70  

     -49    +    -9    =    -58  

     -21    +    -21    =    -42  

     -9    +    -49    =    -58  

For tidiness, printing of 12 lines which failed to find two such factors, was suppressed

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored