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puteri [66]
3 years ago
11

15 points for 1 question??

Mathematics
1 answer:
mestny [16]3 years ago
5 0

Answer:a7b2-ab2  

Final result :

 ab2•(a+1)•(a2-a+1)•(a-1)•(a2+a+1)

Step by step solution :

Step  1  :

Step  2  :

Pulling out like terms :

2.1     Pull out like factors :

  a7b2 - ab2  =   ab2 • (a6 - 1)  

Trying to factor as a Difference of Squares :

2.2      Factoring:  a6 - 1  

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 : 1 is the square of 1

Check :  a6  is the square of  a3  

Factorization is :       (a3 + 1)  •  (a3 - 1)  

Trying to factor as a Sum of Cubes :

2.3      Factoring:  a3 + 1  

Theory : A sum 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 :  1  is the cube of   1  

Check :  a3 is the cube of   a1

Factorization is :

            (a + 1)  •  (a2 - a + 1)  

Trying to factor by splitting the middle term

2.4     Factoring  a2 - a + 1  

The first term is,  a2  its coefficient is  1 .

The middle term is,  -a  its coefficient is  -1 .

The last term, "the constant", is  +1  

Step-1 : Multiply the coefficient of the first term by the constant   1 • 1 = 1  

Step-2 : Find two factors of  1  whose sum equals the coefficient of the middle term, which is   -1 .

     -1    +    -1    =    -2  

     1    +    1    =    2  

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Trying to factor as a Difference of Cubes:

2.5      Factoring:  a3-1  

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 :  1  is the cube of   1  

Check :  a3 is the cube of   a1

Factorization is :

            (a - 1)  •  (a2 + a + 1)  

Trying to factor by splitting the middle term

2.6     Factoring  a2 + a + 1  

The first term is,  a2  its coefficient is  1 .

The middle term is,  +a  its coefficient is  1 .

The last term, "the constant", is  +1  

Step-1 : Multiply the coefficient of the first term by the constant   1 • 1 = 1  

Step-2 : Find two factors of  1  whose sum equals the coefficient of the middle term, which is   1 .

     -1    +    -1    =    -2  

     1    +    1    =    2  

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Final result :

 ab2•(a+1)•(a2-a+1)•(a-1)•(a2+a+1)

Step-by-step explanation:

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