Problem 55 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ (x+y+z)(x-y-z) $$

Step-by-Step Solution

Answer

The simplified form is \(x^2 - y^2 - z^2\).

Step 1: Apply the Distributive Property

To simplify \((x+y+z)(x-y-z)\), we begin by applying the distributive property. This involves multiplying each term in the first parenthesis by each term in the second parenthesis. We'll do them in succession: 1. \(x\) multiplied by each term in \((x-y-z)\): - \(x \cdot x = x^2\), - \(x \cdot -y = -xy\), - \(x \cdot -z = -xz\).2. \(+y\) multiplied by each term in \((x-y-z)\): - \(y \cdot x = yx\), - \(y \cdot -y = -y^2\), - \(y \cdot -z = -yz\).3. \(+z\) multiplied by each term in \((x-y-z)\): - \(z \cdot x = zx\), - \(z \cdot -y = -zy\), - \(z \cdot -z = -z^2\).

Step 2: Combine Like Terms

Next, we need to add all the terms from the expanded expression:\[x^2 - xy - xz + yx - y^2 - yz + zx - zy - z^2.\]Now, combine like terms to simplify:- Combine \(xy\) and \(yx\) terms: - \(-xy + yx = 0\), since they are additive inverses.- Combine \(xz\) and \(zx\) terms: - \(-xz + zx = 0\), since they are additive inverses.- Combine \(yz\) and \(zy\) terms: - \(-yz - zy = 0\), since they are additive inverses.The simplified expression is then:\[x^2 - y^2 - z^2.\]

Key Concepts

Expand Algebraic Expressions

Expanding algebraic expressions is like unpacking a packed suitcase, seeing what you have, and laying it all out. We use the distributive property to manage this process. When we are faced with an expression like \((x+y+z)(x-y-z)\), our goal is to remove the parentheses by distributing every term in the first set to every term in the second set. Simply put:

• Each term in the first group multiplies separately with each term in the second group.
• Ensure you cover all combinations.

For instance, in this exercise, we multiply each term of \((x+y+z)\) with each term of \((x-y-z)\). It is critical to be systematic to ensure no pair is missed. As a tip, always remember to carry the sign (positive or negative) in front of each term to avoid errors in expansion.

Simplification

Simplification is the art of making math expressions less complicated, much like tidying up a desk. After expanding algebraic expressions, we often end up with a lengthy expression full of terms. The purpose of simplification is to make this morass of terms digestible and understandable.First, review all the terms in your expanded expression. Pay close attention to like terms, which make simplification possible. In our task, once expanded, the expression is:\[x^2 - xy - xz + yx - y^2 - yz + zx - zy - z^2\]At this stage, you're simply preparing to combine like terms — it’s like looking for pairs of socks in a pile of laundry.

Combine Like Terms

Combining like terms is akin to grouping similar items together. Imagine sorting colored beads or matching socks. In algebra, like terms are terms with the same variables raised to the same power. They can be grouped together for simplification by either adding or subtracting their coefficients.For example, if two terms both have \(xy\) (like \(-xy\) and \(yx\)), they can be combined. In our exercise, combine terms like \(-xy\) and \(yx\), which equal zero due to their opposite signs. Do this with all similar terms:

• \[ -xy + yx = 0 \]
• \[ -xz + zx = 0 \]
• \[ -yz - zy = 0 \]

This process clears out the expression, leaving only distinct terms: \(x^2 - y^2 - z^2\), which is much neater and easier to interpret.