Factorise (p-a)^3 +(p-b)^3 +(p-c)^3 where p=a+b+c/3
So you have
1/27 [ (-2a+b+c)^3 + (a-2b+c)^3 + (a+b-2c)^3 ]
= -1/9 (2a-b-c)(a-2b+c)(a+b-2c)
To factorize the expression (p-a)^3 +(p-b)^3 +(p-c)^3, we can start by applying the formula for factoring the sum of cubes. The formula states that a^3 + b^3 + c^3 can be factored as (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc).
In this case, let's substitute the given value of p, which is p = (a + b + c)/3. We can rewrite the expression as follows:
[(a + b + c)/3 - a]^3 + [(a + b + c)/3 - b]^3 + [(a + b + c)/3 - c]^3
Now, let's simplify each term within the parentheses:
[(a + b + c)/3 - a]^3 = [(-2a + 2b + 2c)/3]^3 = (-2/3)^3 * [a - b - c]^3 = (-8/27) * [a - b - c]^3
Similarly, we find:
[(a + b + c)/3 - b]^3 = (-8/27) * [b - a - c]^3
[(a + b + c)/3 - c]^3 = (-8/27) * [c - a - b]^3
Therefore, our expression can be rewritten as:
(-8/27) * [a - b - c]^3 + (-8/27) * [b - a - c]^3 + (-8/27) * [c - a - b]^3
Now, we have a common factor of (-8/27) in each term. We can factor this out:
(-8/27) * [(a - b - c)^3 + (b - a - c)^3 + (c - a - b)^3]
Finally, notice that the terms inside the brackets are the same but rearranged due to the properties of addition and subtraction. So, we can rewrite the expression as:
(-8/27) * [(a - b - c)^3 + (a - b - c)^3 + (a - b - c)^3]
Simplifying further, we have:
(-8/27) * [3(a - b - c)^3]
Finally, we can factor out the constant term 3 from the brackets:
-8/27 * 3 * (a - b - c)^3
This can be further simplified to:
-8(a - b - c)^3 / 9
Therefore, the fully factorized form of the expression (p-a)^3 +(p-b)^3 +(p-c)^3 is -8(a - b - c)^3 / 9.