Factor a 3 - 3 + 3a 2 - a. (a - 1)(a + 1)(a + 3) (a2 + 1)(a - 3) (a2 - 3)(a + 1)

Changes made to your input should not affect the solution:

 (1): "a2"   was replaced by   "a^2".  2 more similar replacement(s).

(2): Dot was discarded near "a.(".

Find roots (zeroes) of :       F(a) = a2+1
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  1  and the Trailing Constant is  1. 

 The factor(s) are: 

of the Leading Coefficient :  1
 of the Trailing Constant :  1 

 Let us test

Polynomial Roots Calculator found no rational roots

Factoring:  a2-3 

(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 : 3 is not a square !! 

Ruling : Binomial can not be factored as the difference of two perfect squares.

Multiply  (a+1)  by  (a+1) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (a+1)  and the exponents are :
          1 , as  (a+1)  is the same number as  (a+1)1 
 and   1 , as  (a+1)  is the same number as  (a+1)1 
The product is therefore,  (a+1)(1+1) = (a+1)2 

 Pull out like factors :

   -a10 - a9 + 12a8 + 12a7 - 26a6 - 26a5 - 12a4 - 12a3 + 30a2 + 27a  = 

  -a • (a9 + a8 - 12a7 - 12a6 + 26a5 + 26a4 + 12a3 + 12a2 - 30a - 27) 

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