Question

Factories: a^7b +ab^7.




only a brainlist genius can answer thi by full method

Answer 1

a7b-ab7 

Final result :

ab•(a+b)•(a2-ab+b2)•(a-b)•(a2+ab+b2)

Step by step solution :

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   a7b - ab7  =   ab • (a6 - b6) 

Trying to factor as a Difference of Squares :

 2.2      Factoring:  a6 - b6 

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 :  a6  is the square of  a3 

Check :  b6  is the square of  b3 

Factorization is :       (a3 + b3)  •  (a3 - b3) 

Trying to factor as a Sum of Cubes :

 2.3      Factoring:  a3 + b3 

Theory : A sum of two perfect cubes,  a3 + b3can be factored into  :
             (a+b) • (a2-ab+b2)
Proof  : (a+b) • (a2-ab+b2) = 
    a3-a2b+ab2+ba2-b2a+b3 =
    a3+(a2b-ba2)+(ab2-b2a)+b3=
    a3+0+0+b3=
    a3+b3

Check :  a3 is the cube of   a1

Check :  b3 is the cube of   b1

Factorization is :
             (a + b)  •  (a2 - ab + b2) 

Trying to factor a multi variable polynomial :

 2.4    Factoring    a2 - ab + b2 

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

 Factorization fails

Trying to factor as a Difference of Cubes:

 2.5      Factoring:  a3 - b3 

Theory : A difference of two perfect cubes,  a3 - b3 can be factored into
              (a-b) • (a2 +ab +b2)

Proof :  (a-b)•(a2+ab+b2) =
            a3+a2b+ab2-ba2-b2a-b3 =
            a3+(a2b-ba2)+(ab2-b2a)-b3 =
            a3+0+0+b3 =
            a3+b3

Check :  a3 is the cube of   a1

Check :  b3 is the cube of   b1

Factorization is :
             (a - b)  •  (a2 + ab + b2) 

Trying to factor a multi variable polynomial :

 2.6    Factoring    a2 + ab + b2 

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

 Factorization fails

Final result :

ab•(a+b)•(a2-ab+b2)•(a-b)•(a2+ab+b2)