Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  34a4 -  256

Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  81a⁴-256 

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 :  81  is the square of  9 
Check : 256 is the square of 16
Check :  a⁴  is the square of  a² 

Factorization is :       (9a² + 16)  •  (9a² - 16) 

Polynomial Roots Calculator :

 2.2    Find roots (zeroes) of :       F(a) = 9a² + 16
Polynomial Roots Calculator is a set of methods aimed at finding values of  a  for which   F(a)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  a  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  9  and the Trailing Constant is  16.

 The factor(s) are:

of the Leading Coefficient :  1,3 ,9
 of the Trailing Constant :  1 ,2 ,4 ,8 ,16

 Let us test ....

Note - For tidiness, printing of 25 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Squares :

 2.3      Factoring:  9a² - 16 

Check :  9  is the square of  3 
Check : 16 is the square of 4
Check :  a²  is the square of  a¹ 

Factorization is :       (3a + 4)  •  (3a - 4) 

Final result :

  (9a2 + 16) • (3a + 4) • (3a - 4)