Problem 6 - Algebra 1: Concepts and Skills

Question

Use a special product pattern to find the product. $$ (3 y-1)^{2} $$

Step-by-Step Solution

Answer

The product of \( (3y - 1)^2 \) using the binomial square formula is \( 9y^2 - 6y + 1 \).

Step 1: Identify the Components of the Binomial

The binomial is \( (3y - 1)^2 \). Here, \(a = 3y\) and \(b = 1\).

Step 2: Apply the Binomial Square Formula

The binomial square formula \( (a - b)^2 = a^2 - 2ab + b^2 \) can be applied to \( (3y - 1)^2 \). Substituting the identified components, \(a=3y\) and \(b=1\), into the formula results in: \( (3y)^2 - 2(3y)(1) + 1^2 \).

Step 3: Simplify the Expression

Now, simplify the above expression. \( (3y)^2 = 9y^2 \), \(2(3y)(1) = 6y\), and \(1^2 = 1\). So, the simplified expression is: \( 9y^2 - 6y + 1 \).

Key Concepts

Special Product Patterns

Understanding the concept of special product patterns can make algebraic problems much easier to solve. These patterns are essentially shortcuts that help you quickly find the products of certain types of expressions. They are particularly useful when dealing with polynomials.

One of the most common special product patterns is the binomial square formula. Recognizing these patterns can save you time when you're expanding expressions like \((x + y)^2\) or \((a - b)^2\). If you see a problem like \((3y - 1)^2\), as in this exercise, you can immediately apply this pattern without having to multiply each term individually.

Being familiar with these patterns not only speeds up your work but also helps reduce errors that may occur during manual multiplication. So, recognizing and applying these special patterns is a helpful step toward mastering algebra.

Binomial Square Formula

The binomial square formula is a specific type of special product pattern. It is used when you are squaring a binomial, which is an algebraic expression containing two terms. The formula states that\((a - b)^2\) is equal to a squared minus two times the product of a and b, plus b squared.

Applying this formula to a binomial like \((3y - 1)^2\), you start by identifying a and b from the expression. In this case, a = 3y and b = 1.

Substitute these values into the formula:\((3y - 1)^2 = (3y)^2 - 2(3y)(1) + 1^2\).

This handy formula allows you to find the expanded form of the squared binomial quickly and efficiently, which is crucial in simplifying complex algebraic expressions and solving equations.

Simplifying Expressions

Simplifying expressions is the process of making an equation or algebraic expression as simple as possible. This often involves performing arithmetic operations and combining like terms. When you use the binomial square formula, as we did with \((3y - 1)^2\), you end up with an expression that needs to be simplified.

After substituting the values in the formula, you'll calculate each part:\((3y)^2 = 9y^2\), \(2(3y)(1) = 6y\), and \(1^2 = 1\).

Once you've calculated these, you combine them together to get the simplified expression: \(9y^2 - 6y + 1\).

• Look to perform operations like distributing, combining similar terms, or using arithmetic operations.
• Always check your work to ensure that you've simplified as much as possible.

Simplifying expressions is a fundamental skill in algebra that makes it easier to work with larger and more complex equations. It helps reveal the core components of the expression or equation, making further algebraic operations more straightforward.