1000a^2+27b^4 factorise
Answers
Step-by-step explanation:
STEP
1
:
Equation at the end of step 1
(1000 • (a6)) - 33b3
STEP
2
:
Equation at the end of step
2
:
(23•53a6) - 33b3
STEP
3
:
Trying to factor as a Difference of Squares:
3.1 Factoring: 1000a6-27b3
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 : 1000 is not a square !!
Ruling : Binomial can not be factored as the
difference of two perfect squares
Trying to factor as a Difference of Cubes:
3.2 Factoring: 1000a6-27b3
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 : 1000 is the cube of 10
Check : 27 is the cube of 3
Check : a6 is the cube of a2
Check : b3 is the cube of b1
Factorization is :
(10a2 - 3b) • (100a4 + 30a2b + 9b2)
Trying to factor as a Difference of Squares:
3.3 Factoring: 10a2 - 3b
Check : 10 is not a square !!
Ruling : Binomial can not be factored as the
difference of two perfect squares
Trying to factor a multi variable polynomial :
3.4 Factoring 100a4 + 30a2b + 9b2
Try to factor this multi-variable trinomial using trial and error
Factorization fails
Final result :
(10a2 - 3b) • (100a4 + 30a2b + 9b2)