Proving Algebraic Identities: A Closer Look at a b c a(b c) - bc(a c) - ca(b a) - abc

In this article, we delve into the proof of the algebraic identity a b c a(b c) - bc(a c) - ca(b a) - abc. We will follow a detailed step-by-step process, leveraging polynomial properties and symmetry for clarity.

Step 1: Expanding the Left-Hand Side

The left-hand side of the given identity is a b c.

Step 2: Expanding the Right-Hand Side

To start with, we expand the right-hand side, which is a(b c) - bc(a c) - ca(b a) - abc.

Substep 2.1: Expanding a(b c)

First, let's expand a(b c) to get:

[ a(b c) ab ac ]

Substep 2.2: Expanding -bc(a c)

Next, we expand -bc(a c) to get:

[ -bc(a c) -b^2c - bc^2 ]

Substep 2.3: Expanding -ca(b a)

Now, we expand -ca(b a) to get:

[ -ca(b a) -acb - ca^2 ]

Substep 2.4: Combining All Terms

Finally, we combine all the terms, including -abc, to get the right-hand side as:

[ ab ac - b^2c - bc^2 - acb - ca^2 - abc ]

The above expression simplifies to:

[ ab ac - b^2c - bc^2 - acb - ca^2 - abc ]
ab ac - b^2c - bc^2 - abc ]

Step 3: Simplifying and Comparing Both Sides

Let's now compare the terms on both sides.

For the left-hand side, we have:

[ a b c ]

For the right-hand side, we have:

[ ab ac - b^2c - bc^2 - abc ]

Notice that both sides contain the same terms, just arranged differently.

Conclusion

As we can see, the simplified form of the right-hand side matches the left-hand side, proving the identity:

[ a b c a(b c) - bc(a c) - ca(b a) - abc ]

Additional Observations

By substituting ( b -a ) on the right-hand side, we find that the expression evaluates to zero. This implies that ( ab ) is a factor of the right-hand side. Similarly, by symmetry, ( bc ) and ( ca ) are also factors.

Considering the polynomials ( f(x) a(b x) - bc(a x) - ca(b x) - abc ) and ( g(x) a(b x) - bc(a x) - ca(b x) - abc ), both coincide at ( x -a ) and ( x -b ). Therefore, we can conclude that both are identical polynomials.

The final observation that both polynomials are of degree 3 further confirms our proof.

Keywords: algebraic identities, polynomial properties, identity proof