Factorise:-
27p^3-1/216-9/2p^2+1/4p
To factorize the given expression: 27p^3 - 1/216 - 9/2p^2 + 1/4p, we can follow these steps:
Step 1: Look for common factors, if any. In this case, we don't have any common factors.
Step 2: Group the terms if possible. For this expression, we can group the first two terms together and the last two terms together:
(27p^3 - 1/216) - (9/2p^2 - 1/4p)
Step 3: Factor out the greatest common factor (GCF) from each group. In the first group, the GCF is 1/216, while in the second group, the GCF is 1/4p:
(1/216)(27p^3 - 1) - (1/4p)(9p - 1)
Step 4: Now, we need to factor the difference of cubes in the first group. The difference of cubes formula is given by a^3 - b^3 = (a - b)(a^2 + ab + b^2). In this case, a = 3p and b = 1. We can apply the formula:
(1/216)[(3p)^3 - 1^3] - (1/4p)(9p - 1)
Simplifying further:
(1/216)((3p - 1)(9p^2 + 3p + 1)) - (1/4p)(9p - 1)
Therefore, the factored form of the given expression is:
(1/216)((3p - 1)(9p^2 + 3p + 1)) - (1/4p)(9p - 1)
To factorize the given expression 27p^3 - 1/216 - 9/2p^2 + 1/4p, we can first group the terms together:
(27p^3 - 9/2p^2) - (1/216 - 1/4p)
Now, let's factor out the common terms from each group:
(3p^2) * (9p - 3/2) - (1/216 - 1/4p)
Next, let's simplify the expression within the parentheses:
(3p^2) * (18p/2 - 3/2) - (1/216 - 1/4p)
Simplifying further:
(3p^2) * (9p - 3/2) - (1/216 - 54p/216)
Now, let's find a common denominator for the fractions:
(3p^2) * (9p - 3/2) - ((1 - 54p)/216)
Combining the fractions:
(3p^2) * (9p - 3/2) - (1 - 54p)/216
Now, let's find a common factor for the numerator of the second fraction:
(3p^2) * (9p - 3/2) - (1 - 6 * 9p)/(6 * 36)
Simplifying further:
(3p^2) * (9p - 3/2) - (1 - 54p)/(216)
Now, we can write the factored form of the expression:
(3p^2) * (9p - 3/2) - (1 - 54p)/(216)