Simplifying and Factoring the Expression {cb-a}^2 - {a-b-c}^2

To understand how to factor the expression {cb-a}^2 - {a-b-c}^2, we can utilize a well-known algebraic identity, the difference of squares. This identity states that x^2 - y^2 (x - y)(x y). Let's break down the process step by step.

Step-by-Step Breakdown

Step 1: Identify x and y in the expression.

Let x cb-a and y a-b-c. Then, we can rewrite the given expression as follows:

{cb-a}^2 - {a-b-c}^2 x^2 - y^2.

Step 2: Apply the difference of squares formula.

Using the formula, we get:

x^2 - y^2 (x - y)(x y).

Step 3: Calculate x - y and x y.

x - y:

x - y (cb - a) - (a - b - c)

Simplifying this, we get:

x - y cb - a - a b c 2c 2b - 2a 2(c b - a).

x y:

x y (cb - a) (a - b - c) cb - a a - b - c cb - b - c.

Step 4: Substitute x - y and x y back into the formula.

x^2 - y^2 (2(c b - a))(cb - b - c).

Step 5: Evaluate the expression.

[cb - a]^2 - [a - b - c]^2 (2(c b - a))(cb - b - c).

However, we notice that {cb-a} -{a-b-c}, therefore {cb-a}^2 {a-b-c}^2. Thus, substituting back:

{cb-a}^2 - {a-b-c}^2 {a-b-c}^2 - {a-b-c}^2 0.

Hence, the expression simplifies to 0, and we do not need further factorization.

Alternative Approach

Another approach is to use the identity directly:

{cb-a}^2 - {a-b-c}^2

By recognizing that the two terms are squared and have the same value when we use the identity, we get:

{cb-a}^2 - {a-b-c}^2 (cb - a)^2 - (a - b - c)^2 (a - b - c)^2 - (a - b - c)^2 0.

This further confirms that the expression is identically zero for all values of a, b, and c.

Conclusion

The expression {cb-a}^2 - {a-b-c}^2 is indeed always equal to 0, and thus no further factorization is required. This is a result of the symmetry and the fact that squaring a term and its negation results in the same value.

Understanding these concepts in algebra is crucial for solving a variety of problems and is a fundamental skill for any student or professional working in mathematics or related fields.

Key Concepts: The difference of squares, x^2 - y^2 (x - y)(x y), and the property of squared terms.