Solution - Factoring multivariable polynomials

Step  1  :

Equation at the end of step  1  :

  ((81•(x4))+(16•(y4)))-((23•32x2)•y2)
 Step  2  :

Equation at the end of step  2  :

  ((81 • (x4)) +  24y4) -  (23•32x2y2)
 Step  3  :

Equation at the end of step  3  :

  (34x4 +  24y4) -  (23•32x2y2)

Step  4  :

Trying to factor a multi variable polynomial :

 4.1    Factoring    81x⁴ - 72x²y² + 16y⁴ 

Try to factor this multi-variable trinomial using trial and error 

 Found a factorization  :  (9x² - 4y²)•(9x² - 4y²)

Detecting a perfect square :

 4.2    81x⁴  -72x²y²  +16y⁴  is a perfect square 

 It factors into  (9x²-4y²)•(9x²-4y²)
which is another way of writing  (9x²-4y²)²

How to recognize a perfect square trinomial:  

 • It has three terms  

 • Two of its terms are perfect squares themselves  

 • The remaining term is twice the product of the square roots of the other two terms

Trying to factor as a Difference of Squares :

 4.3      Factoring:  9x²-4y² 

Put the exponent aside, try to factor  9x²-4y² 

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 :  9  is the square of  3 
Check : 4 is the square of 2
Check :  x²  is the square of  x¹ 

Check :  y²  is the square of  y¹ 

Factorization is :       (3x + 2y)  •  (3x - 2y) 

Raise to the exponent which was put aside
Factorization becomes :   (3x + 2y)²   •  (3x - 2y)²  

Final result :

  (3x + 2y)2 • (3x - 2y)2