Problem 16 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. \((3 b-2)^{2}\)

Step-by-Step Solution

Answer

The result is \(9b^2 - 12b + 4\).

Step 1: Identify the Expression to Expand

The expression given is \((3b - 2)^2\). This indicates we need to multiply \((3b - 2)\) by itself.

Step 2: Use the Distributive Property

To expand \((3b - 2)^2\), we prepare to multiply each term in the first binomial by each term in the second binomial: \((3b - 2)(3b - 2)\).

Step 3: Multiply Each Term

First, multiply the first terms: \(3b \times 3b = 9b^2\). Then, multiply the outer terms: \(3b \times -2 = -6b\). Next, multiply the inner terms: \(-2 \times 3b = -6b\). Finally, multiply the last terms: \(-2 \times -2 = 4\).

Step 4: Combine Like Terms

The expanded form after distribution gives \(9b^2 - 6b - 6b + 4\). Combine the like terms \(-6b\) and \(-6b\) to get \(-12b\). The final expression is \(9b^2 - 12b + 4\).

Key Concepts

Distributive Property

When expanding expressions like \((3b - 2)^2\), we use a powerful mathematical tool called the distributive property. This property allows us to break down larger expressions into simpler parts, making them much easier to handle. To apply the distributive property, we multiply each term in one expression by each term in another. In the case of a squared binomial, such as \((3b - 2)^2\), it requires multiplying the binomial by itself: \((3b - 2)(3b - 2)\). This multiplication is not straightforward compared to simple coefficient multiplication but follows a consistent pattern, which helps in expanding the problem efficiently. By ensuring that each term in one binomial multiplies with every term in the other, all possible products are found, thus leveraging the full might of the distributive property.

Binomial Multiplication

Binomial multiplication involves multiplying two binomial expressions, which are algebraic expressions containing two terms each. In our example, we have \((3b - 2)(3b - 2)\). To perform this multiplication, we follow a structured approach: First, multiply the first terms of both binomials: \(3b \times 3b = 9b^2\). Next, multiply the outer terms: \(3b \times -2 = -6b\). Then, multiply the inner terms: \(-2 \times 3b = -6b\). Lastly, multiply the last terms: \(-2 \times -2 = 4\). This structured approach ensures that we do not miss any combinations and capture all components of the expanded polynomial.

Combining Like Terms

Combining like terms is the process used in algebra to simplify expressions by consolidating terms with the same variables and exponents. After expanding a binomial, like in our example, we initially have an expression such as \(9b^2 - 6b - 6b + 4\). To simplify this, we identify terms with the same variables and combine them. Here, the terms \(-6b\) and \(-6b\) are both like terms because they have the same variable, \(b\), raised to the same power. Combining these yields \(-12b\). Thus, the expression simplifies to \(9b^2 - 12b + 4\). This process helps in reducing the complexity of the expression, allowing for easier interpretation and further mathematical operations.