Factorize a3 +b3 +c3 −3abc

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:

• x = a
• y = b

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:

• x = a + b
• y = c

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)