Problem 31 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ (x+2 y)(3 x-y) $$

Step-by-Step Solution

Answer

\(3x^2 + 5xy - 2y^2\)

Step 1: Distribute the First Term

First, distribute the term \(x\) from \((x + 2y)\) to each term in \((3x - y)\). This yields:\[x \cdot 3x + x \cdot (-y) = 3x^2 - xy\]

Step 2: Distribute the Second Term

Next, distribute the term \(2y\) from \((x + 2y)\) to each term in \((3x - y)\). This gives us:\[2y \cdot 3x + 2y \cdot (-y) = 6xy - 2y^2\]

Step 3: Combine Like Terms

Now, combine all terms obtained from the previous distribution operations:\[3x^2 - xy + 6xy - 2y^2\]Combine the like terms, specifically the \(-xy + 6xy\):\[3x^2 + 5xy - 2y^2\]

Step 4: Write the Final Expression

Finally, write down the simplified expression, which includes all combined terms:\[3x^2 + 5xy - 2y^2\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us multiply expressions. It's particularly useful when dealing with polynomial multiplication. This property states that you can multiply a single term by each term inside the brackets separately. In simpler words, when you have

• The expression: - \(a(b + c)\)
• You can distribute - \(a \), to each term: - \(ab + ac\)

Applying this to the expression \( (x + 2y)(3x - y)\), we start by distributing both \(x\) and \(2y\) to each term in the second parenthesis.

• First: Multiply \(x\) by each term: - \(x \cdot \ 3x = 3x^2\) - \(x \cdot \ (-y) = -xy\)
• Next: Multiply \(2y\) by each term: - \(2y \cdot \ 3x = 6xy\) - \(2y \cdot \ (-y) = -2y^2\)

This step-by-step distribution helps break down the multiplication process into manageable chunks. Once you understand this property, tackling more complex multiplications becomes less daunting.

Combine Like Terms

Combining like terms is a vital step in simplifying polynomial expressions. Like terms are terms that have the same variables raised to the same powers, even if the coefficients are different.

Consider the expression: \(3x^2 - xy + 6xy - 2y^2\).

• Identifying the like terms here involves looking for terms with the same variable parts: - \(-xy\) and \(6xy\) are like terms because they both contain the variables \(x\) and \(y\).
• To combine these, simply add their coefficients together: - \(-xy + 6xy = 5xy\)

Once combined, your expression should read: \(3x^2 + 5xy - 2y^2\).
Combining like terms simplifies the expression, making it easier to read and understand.

Expression Simplification

Expression simplification involves making an algebraic expression more straightforward, combining like terms, and simplifying it to its most reduced form.
It's crucial for solving equations and making expressions easier to work with in future calculations. By using the distributive property and combining like terms, you break down complex expressions into simple parts.
In our example,

• We started with a multiplication problem: \( (x + 2y)(3x - y)\)
• Through the steps of applying the distributive property and combining like terms, we found the simplified form: - \(3x^2 + 5xy - 2y^2\)

Simplification is not just a mathematical procedure; it represents a critical step towards understanding relationships in algebra and serves as the foundation for solving more advanced problems.