factorise
a) m^3-27n^3
b) 64a^3b^4+216a^6b
c) (a+b)-a-b
1.
from A^3 - B^3 = (A-B)(A^2 + AB + B^2)
m^3 - 27n^3
= (m-3n)(m^2 + 3mn + 9n^2)
2.
64a^3b^4+216a^6b
= 8a^3b(8b^3 + 27a^3)
= 8a^3 b(2b+3a)(4b^2 - 6ab + 9a^2)
3. (a+b) - a - b
= a+b-a-b
= 0
To factorize these expressions, we need to find common factors and use algebraic identities.
a) To factorize m^3 - 27n^3, we can recognize this as a difference of cubes. The formula for the difference of cubes is a^3 - b^3 = (a - b)(a^2 + ab + b^2).
So, we can rewrite our expression as (m)^3 - (3n)^3. Now we can apply the formula to factorize it:
m^3 - 27n^3 = (m - 3n)(m^2 + 3mn + 9n^2)
b) To factorize 64a^3b^4 + 216a^6b, we can first notice that both terms have a common factor of 8ab. We can factor that out:
64a^3b^4 + 216a^6b = 8ab(8a^2b^3 + 27a^5)
Now, we can see that the remaining expression is a sum of cubes. The formula for the sum of cubes is a^3 + b^3 = (a + b)(a^2 - ab + b^2). So, we can apply the formula to factorize the remaining term:
8a^2b^3 + 27a^5 = (2ab + 3a^2)(4a^2b^2 - 6ab + 9a^3)
Combining both factors, we get the final factorization:
64a^3b^4 + 216a^6b = 8ab(2ab + 3a^2)(4a^2b^2 - 6ab + 9a^3)
c) To factorize (a + b) - a - b, we can simplify it by combining like terms:
(a + b) - a - b = a + b - a - b = (a - a) + (b - b) = 0
Therefore, the factorization of (a + b) - a - b is 0.