Question

Factorise x^4y^4 - 256 z^4

Answer 1

Step  1  :

Equation at the end of step  1  :

((x4) • (y4)) - 28z4

Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  x4y4-256z4 

Theory : A difference of two perfect squares, A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A2 - AB + BA - B2 =
         A2 - AB + AB - B2 = 
         A2 - B2

Note :  AB = BA is the commutative property of multiplication. 

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 256 is the square of 16
Check :  x4  is the square of  x2 

Check :  y4  is the square of  y2 

Check :  z4  is the square of  z2 

Factorization is :       (x2y2 + 16z2)  •  (x2y2 - 16z2) 

Trying to factor as a Difference of Squares :

 2.2      Factoring:  x2y2 - 16z2 

Check : 16 is the square of 4
Check :  x2  is the square of  x1 

Check :  y2  is the square of  y1 

Check :  z2  is the square of  z1 

Factorization is :       (xy + 4z)  •  (xy - 4z) 

Final result :

(x2y2 + 16z2) • (xy + 4z) • (xy - 4z)