Proving the Algebraic Identity: a^2 - 2abb^2 a - b^2

In algebra, proving certain identities is crucial for understanding deeper mathematical concepts. One such identity is a^2 - 2abb^2 a - b^2. This article will guide you through the process of proving this identity using various algebraic methods, ensuring clarity and completeness.

Introduction to the Identity

The given identity is: a^2 - 2abb^2 a - b^2. This identity holds under the commutative and associative properties, meaning that in any algebraic structure where these properties are true (like the real numbers), the identity is valid.

Proof Method 1

We will start by expanding and simplifying the left-hand side (L.H.S) to show that it is equal to the right-hand side (R.H.S).

Step 1: Start with the L.H.S. Step 2: Expand the terms and regroup them. Step 3: Simplify the expression. Step 4: Show that it equals the R.H.S.

Method 1:

L.H.S. a^2 - 2ab b^2
a^2 - ab - ab b^2
a^2 - ab - ab b^2
a(a - b) - b(a - b)
(a - b)(a - b)
a - b^2

R.H.S. a - b^2

Hence, a^2 - 2ab b^2 a - b^2.

Proof Method 2

We will now prove the identity from the other side (R.H.S. to L.H.S.), showing that it is also consistent.

Step 1: Start with the R.H.S. Step 2: Expand and regroup the terms. Step 3: Simplify the expression. Step 4: Show that it equals the L.H.S.

Method 2:

R.H.S. a - b^2
a - b(a - b)
aa - ab - ba b^2
a^2 - ab - ab b^2
a^2 - 2ab b^2

L.H.S. a^2 - 2ab b^2

Hence, a^2 - 2ab b^2 a - b^2.

Validation for Real Numbers

The given identity is particularly interesting when considered over the real numbers. We can use polynomial identities to validate this.

Step 1: Recognize that x^2 - 2x(1-x) - (1-x)^2 is a polynomial of degree 2. Step 2: Determine the number of real roots. Step 3: Use the roots to verify the identity.

As x^2 - 2x(1-x) - (1-x)^2 has a degree 2, it can have at most 2 real roots. However, there are at least 3 roots 1, -1, 0. Thus, by the fundamental theorem of algebra, x^2 - 2x(1-x) - (1-x)^2 0.

For nonzero b, let x a/b. Substituting this into the polynomial gives:

a/b^2 - 2a/b(1 - a/b) (a/b) - 1^2
Multiplying both sides by b^2 gives:
a^2 - 2ab b^2 a - b^2

For b 0 and any a, the identity still holds as both sides equal a - 0 a.

Conclusion

The identity a^2 - 2abb^2 a - b^2 is valid under the commutative and associative properties. This method of proof demonstrates the power of algebraic manipulation and polynomial properties in verifying mathematical identities. By following the steps outlined, we can ensure a thorough understanding of the proof and its implications.