Problem 54 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \((x-3 y)^{2}\)

Step-by-Step Solution

Answer

The expression \((x-3y)^2\) simplifies to \(x^2 - 6xy + 9y^2\).

Step 1: Identify the Special Product Formula

The expression \((x-3y)^2\) is a square of a binomial. The special product formula for squaring a binomial \((a - b)^2\) is \(a^2 - 2ab + b^2\). Identify \(a\) and \(b\) in the expression.

Step 2: Assign Values to Variables

In the expression \((x-3y)^2\), assign \(a = x\) and \(b = 3y\). Substitute these into the formula \(a^2 - 2ab + b^2\).

Step 3: Apply the Formula

Substitute \(a = x\) and \(b = 3y\) into the formula: \(a^2 - 2ab + b^2 = x^2 - 2(x)(3y) + (3y)^2\).

Step 4: Calculate Each Term

Calculate each term of the expression: 1. \(x^2 = x^2\).2. \(-2(x)(3y) = -6xy\).3. \((3y)^2 = 9y^2\).

Step 5: Combine the Terms

Combine all the terms to get the simplified expression: \(x^2 - 6xy + 9y^2\).

Key Concepts

Square of a Binomial

When you encounter an expression like \((x - 3y)^2\), you're working with the **square of a binomial**. This means the expression is a binomial raised to the second power. Binomials are algebraic expressions containing exactly two terms, like \(x\) and \(-3y\) in our example.

The special product formula comes in handy here. For the square of a binomial \((a - b)^2\), the formula is:

• \(a^2 - 2ab + b^2\)

This formula helps streamline multiplication processes. Instead of expanding \((x - 3y)(x - 3y)\), we can directly apply this formula.

Identifying each component in the binomial is crucial. That way, you can accurately substitute into the formula without making mistakes.

Algebraic Expressions

Understanding **algebraic expressions** is key to grasping operations like squaring a binomial. An algebraic expression is simply a combination of numbers, variables, and operations (like addition and subtraction). It may contain constants and coefficients as well.

In our example \((x - 3y)^2\), the expression involves:

• A variable \(x\)
• A variable \(y\) with a coefficient of \(-3\)

The square operation is an example of working with these expressions to create a new product. By understanding each part of the expression, such as terms, coefficients, and variables, you set yourself up to manage more complex algebraic operations in the future.

Recognize that handling these expressions properly means knowing how to manipulate them using established mathematical rules and formulas.

Expression Simplification

Simplifying an expression like \(x^2 - 6xy + 9y^2\) involves consolidating terms to make the expression as straightforward as possible. **Expression simplification** is about rewriting an expression in a cleaner, more concise form while ensuring its value remains the same.

To simplify the result of squaring the binomial \((x - 3y)^2\), we calculated each term separately:

• \(x^2\) stayed \(x^2\)
• \(-2(x)(3y)\) simplified to \(-6xy\)
• \((3y)^2\) became \(9y^2\)

Finally, we combined these to form the expression \(x^2 - 6xy + 9y^2\).

It's important to note that simplification doesn't change the meaning or value of the expression; it only makes the expression easier to read and work with. Keeping practice with simplification sharpens algebraic skills considerably.