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)
Explanation: