Factorise..8(l+m)^3 -27(l-m)^3
again, a difference of cubes
(2(l+m)-3(l-m))(4(l+m)^2 + 2(l_m)*3(l-m) + 9(l-m)^2)
(-l+5m)(19l^2-10lm+7m^2)
difference of cube pattern
8(l+m)^3 -27(l-m)^3
= (2(l+m) - 3(l-m) )(4(l+m)^2 + 6(l+m)(l-m) + 9(l-m)^2)
= (-l + 5m)(4l^2 + 8lm + 4m^2 + 6l^2 - 6m^2 + 9l^2 - 18lm + 9m^2)
= (5m - 6)(19l^2 - 10lm + 7m^2)
To factorize the expression 8(l+m)^3 - 27(l-m)^3, we can use the formula for factoring the difference of cubes, which states that:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
In this case, let's rewrite the expression as:
8(l+m)^3 - 27(l-m)^3 = (2)^3(l+m)^3 - (3)^3(l-m)^3
Using the formula, we have:
= [2(l+m) - 3(l-m)] [(2(l+m))^2 + (2(l+m))(3(l-m)) + (3(l-m))^2]
Simplifying the expressions, we have:
= (2l + 2m - 3l + 3m) (4(l^2 + 2lm + m^2) + 6(l^2 - lm - lm + m^2) + 9(l^2 - 2lm + m^2))
= (-l + 5m) (4l^2 + 8lm + 4m^2 + 6l^2 - 12lm + 6m^2 + 9l^2 - 18lm + 9m^2)
= (-l + 5m) (19l^2 - 22lm + 19m^2)
Therefore, the expression 8(l+m)^3 - 27(l-m)^3 is factorized as (-l + 5m)(19l^2 - 22lm + 19m^2).
To factorize the given expression 8(l+m)^3 - 27(l-m)^3, let's first recognize that it is in the form of a difference of cubes. The formula for factoring a difference of cubes is:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
In our given expression, we have (l+m)^3 and (l-m)^3, which can be seen as the cubes of terms.
So, we can rewrite the expression as:
8(l+m)^3 - 27(l-m)^3 = (2(l+m) - 3(l-m))((2(l+m))^2 + (2(l+m))(3(l-m)) + (3(l-m))^2)
Now, let's simplify the terms:
2(l+m) = 2l + 2m
3(l-m) = 3l - 3m
Replacing these values, our factorized form becomes:
(2l + 2m - 3l + 3m)((2l + 2m)^2 + (2l + 2m)(3l - 3m) + (3l - 3m)^2)
Simplifying further:
(-l + 5m)(4l^2 + 4lm + 4lm + 4m^2 + 9l^2 - 9lm + 9lm - 9m^2)
Combining like terms:
(-l + 5m)(13l^2 + 13m^2)
Thus, the factorized form of 8(l+m)^3 - 27(l-m)^3 is (-l + 5m)(13l^2 + 13m^2).