Factorize a3 +b3 +c3 −3abc
Last Updated :
06 Feb, 2024
In this article, we’ll walk you through the step-by-step process of factoring a3 +b3 +c3 −3abc.
Answer: Factorizing a3 +b3 +c3 −3abc will give us (a + b + c)(a2 + b2 + c2 – ac – bc- ab).
Solution:
The expression we want to factorize is a3 +b3 +c3 −3abc.
To simplify it, we’ll use the identity:
(x + y)3 = x3 + y3+ 3xy(x + y)
In our case, we can see that:
So, we have:
a3 + b3 + c3 – 3abc = (a + b)3 + c3– 3abc
Now, let’s factorize the expression (a + b)3 + c.
Now, let’s factorize the expression (a + b)3 + c3 – 3abc using the identity:
(x + y)3 = x3 + y3 + 3xy(x + y)
In our case:
So, we can rewrite the expression as:
(a + b)3 + c3 – 3abc = (a + b + c)((a + b)2 – (a + b)c + c2) – 3abc
Now, let’s simplify the expression further:
- (a + b)2 = a2 + 2ab + b2
- (a + b)c = ac + bc
So, we have:
(a + b)3+ c3 – 3abc = (a + b + c)((a2 + 2ab + b2) – (ac + bc) + c2) – 3abc
Now, let’s simplify the terms inside the brackets:
- a2+ 2ab + b2 – (ac + bc) + c2
This simplifies to:
a2 + 2ab + b2 – ac – bc + c2
So, the fully factored form of the polynomial a3 +b3 +c3 −3abc is:
(a + b + c)(a2 + b2 + c2 – ac – bc- ab)
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