Simplifying the Algebraic Expression

The problem asks us to simplify the expression $$(x - y + z)^2 - (x - y - z)^2$$. This expression has the form $$a^2 - b^2$$, which is a difference of squares. We can use the algebraic identity: $$a^2 - b^2 = (a + b)(a - b)$$.

Applying the Difference of Squares Identity

Let's identify 'a' and 'b' in our expression:

• Let $$a = (x - y + z)$$
• Let $$b = (x - y - z)$$

Now, we need to find $$(a + b)$$ and $$(a - b)$$.

Step 1: Calculate (a + b)

$$(a + b) = (x - y + z) + (x - y - z)$$

Remove the parentheses:

$$(a + b) = x - y + z + x - y - z$$

Combine like terms ($$x$$, $$y$$, and $$z$$ terms):

$$(a + b) = (x + x) + (-y - y) + (z - z)$$

$$(a + b) = 2x - 2y + 0$$

So, $$(a + b) = 2x - 2y$$.

Step 2: Calculate (a - b)

$$(a - b) = (x - y + z) - (x - y - z)$$

Remove the parentheses. Remember to distribute the negative sign to each term inside the second set of parentheses:

$$(a - b) = x - y + z - x + y + z$$

Combine like terms ($$x$$, $$y$$, and $$z$$ terms):

$$(a - b) = (x - x) + (-y + y) + (z + z)$$

$$(a - b) = 0 + 0 + 2z$$

So, $$(a - b) = 2z$$.

Step 3: Multiply (a + b) by (a - b)

Now, we multiply the results from Step 1 and Step 2:

$$(a + b)(a - b) = (2x - 2y)(2z)$$

Distribute $$2z$$ to each term inside the first parentheses:

$$(2x - 2y)(2z) = (2x)(2z) - (2y)(2z)$$

Perform the multiplication:

$$(2x)(2z) - (2y)(2z) = 4xz - 4yz$$

Conclusion

The simplified form of the expression $$(x - y + z)^2 - (x - y - z)^2$$ is $$4xz - 4yz$$.

Let's compare this result with the given options:

• Option 1: $$2xz + 2yz$$
• Option 2: $$4xz + 4yz$$
• Option 3: $$4yz - 4xz$$
• Option 4: $$4xz - 4yz$$

Our simplified expression, $$4xz - 4yz$$, matches Option 4.

Revision Table: Simplifying Algebraic Expressions

Review of the key steps involved in simplifying expressions like this:

Additional Information: Algebraic Identities

Algebraic identities are equations that are true for all values of the variables involved. The difference of squares is one common identity. Others include:

• Square of a sum: $$(a + b)^2 = a^2 + 2ab + b^2$$
• Square of a difference: $$(a - b)^2 = a^2 - 2ab + b^2$$
• Sum of cubes: $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$
• Difference of cubes: $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$

Using these identities can significantly simplify complex algebraic expressions and equations. In this specific problem, recognizing the difference of squares form made the simplification much faster than expanding each squared trinomial separately.