Factor this expression completely, 8y^3 - 27

(2y - 3) • (4y2 + 6y + 9)

Step-by-step explanation:

Step by step solution :

Step  1  :

Equation at the end of step  1  :

 23y3 -  27

Step  2  :

Trying to factor as a Difference of Cubes:

2.1      Factoring:  8y3-27  

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 :  8  is the cube of  2  

Check :  27  is the cube of   3  

Check :  y3 is the cube of   y1

Factorization is :

            (2y - 3)  •  (4y2 + 6y + 9)  

Trying to factor by splitting the middle term

2.2     Factoring  4y2 + 6y + 9  

The first term is,  4y2  its coefficient is  4 .

The middle term is,  +6y  its coefficient is  6 .

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

Step-1 : Multiply the coefficient of the first term by the constant   4 • 9 = 36  

Step-2 : Find two factors of  36  whose sum equals the coefficient of the middle term, which is   6 .

     -36    +    -1    =    -37  

     -18    +    -2    =    -20  

     -12    +    -3    =    -15  

     -9    +    -4    =    -13  

     -6    +    -6    =    -12  

     -4    +    -9    =    -13  

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

Final result :

 (2y - 3) • (4y2 + 6y + 9)

25

step-by-step explanation:

step-by-step explanation: