Difference of square

Question

How to  factor x^6 - 64 using difference of squares. The answer is (x-2)(x+2)(x^2+2x+4)(x^2-2x+4)

Answer 1

Hi Harleen,

X⁶ - 64  is a difference of two squares 

 

(X³) ² - 8² = (X³ + 8)(X³ -8)

 

(X³ + 8) = (X³ + 2³) = (X + 2)(X² - 2X +4)       This is the sum of two cubes

(X³  - 8) = (X³   - 2³) = (X  - 2)(X² + 2X +4)       This is the difference of two cubes

Plug these values  into the first equation above to get

(X⁶ + 64)= (X³) ² - 8²

              = (X³ + 8)(X³ -8)

              =(X + 2)(X² - 2X +4)(X - 2)(X² + 2X +4)  

 

 

If you forgot the above identity you can use synthetic division or long division, rewrite the expression 
(X³ + 8) = (X³ + 2³) = X³ + 0X² + 0X + 8

 

               X² - 2X + 4

              --------------------------------

(X + 2) |  X³  +  0X² + 0X + 8

----------

             -(X³ + 2X²)

             -------------

                      - 2X² + 0X

                   - (-2X²  - 4X)

                    ---------------

                                  4X + 8

                                -(4X + 8)

                                 ----------