Algebraic Identity of x³+y³+z³-3xyz.

Suppose, students are sitting in a mathematics class.

The teacher enters in the class. He has to discuss an important identity.

He tells one of his students to write $3$ numbers on the board.

The three numbers should be such that their sum is $0$.

A student writes these numbers on the board.

The teacher asks the student to find the sum of cubes of the numbers.

Then, he tells the student to find thrice of the product of the numbers.

The student finds both the results are equal.

Then some more students try this process and find equal results.

The teacher then tells them it is the application of an important algebraic identity.

Here, we can see that since the sum of $x,y,z$ is zero, RHS also becomes zero.

So, let’s now try to verify whether this identity always holds true.

Let’s consider our algebraic identity.

Let’s first consider the RHS of the identity.

Let’s first simplify the RHS.

We see that the negative terms cancel some positive terms

By simplifying this again, we get this which is equal to the RHS side.

Thus, we can say that the identity holds true algebraically.

Let’s again try to verify this identity numerically.

Suppose we take $x=2$ and $y=3$ and $z=4$ and put the value in above identity.

By putting the values of $x,y,z$ we get the result.

By putting the value of $x,y,z$ in RHS we get the result,

Again, since LHS = RHS, we can say that the identity is numerically true, as well.

This identity has many applications in mathematics one of them is factorisation of polynomials.

Suppose we take a polynomial of $3$ variables $a,b,c$

We can also write this identity in another way,

Let us consider variables $a,b,c$ in the form of $x,y,z$

Now, we will substitute these values in our identity.

By taking $RHS$ in the given identity,

By putting the values of $x,y,z$ in the form of $a,b,c$ variable

Thus, we have factorized the polynomial using our algebraic identity.

Revision

We have an important identity regarding cubes of three numbers.

If $x+y+z=0$ the identity becomes,

We can use these identities to factorize algebraic expressions.

The End