Question

25a^4 - 19a^ 2 b^2 + 9b^4
solve this factorisation.

Answer 1

25a4-30a2b+9b2 

Final result :

(5a2 - 3b)2

Step by step solution :

Step  1  :

Equation at the end of step  1  :

((25•(a4))-((30•(a2))•b))+32b2

Step  2  :

Equation at the end of step  2  :

((25 • (a4)) - ((2•3•5a2) • b)) + 32b2

Step  3  :

Equation at the end of step  3  :

(52a4 - (2•3•5a2b)) + 32b2

Step  4  :

Trying to factor a multi variable polynomial :

 4.1    Factoring    25a4 - 30a2b + 9b2 

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

 Found a factorization  :  (5a2 - 3b)•(5a2 - 3b)

Detecting a perfect square :

 4.2    25a4  -30a2b  +9b2  is a perfect square 

 It factors into  (5a2-3b)•(5a2-3b)
which is another way of writing  (5a2-3b)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:  5a2-3b 

Put the exponent aside, try to factor  5a2-3b 

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 :  5  is not a square !! 

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

Final result :

(5a2 - 3b)2