Problem 29 - College Algebra

Question

For the following exercises, expand the binomial. $$(3 y-6)^{2}$$

Step-by-Step Solution

Answer

(3y-6)^2 = 9y^2 - 36y + 36

Step 1: Understand the Binomial Theorem

The binomial theorem states that \((a + b)^n\) can be expanded into a series of terms based on the powers of \(a\) and \(b\). For this exercise, we will expand using \((a - b)^2\) where \(a = 3y\) and \(b = 6\).

Step 2: Apply the Formula for Squaring a Binomial

The formula for squaring a binomial \((a - b)^2\) is \(a^2 - 2ab + b^2\). We'll use this formula to expand \((3y - 6)^2\) by substituting \(a = 3y\) and \(b = 6\).

Step 3: Calculate Each Term in the Expansion

First, calculate \(a^2 = (3y)^2 = 9y^2\). Next, calculate \(-2ab = -2(3y)(6) = -36y\). Finally, calculate \(b^2 = 6^2 = 36\).

Step 4: Combine the Terms

Combine the calculated terms into the expanded form: \(9y^2 - 36y + 36\).

Step 5: Verify the Calculation

Review each calculation to ensure accuracy. Confirm that \(9y^2\), \(-36y\), and \(+36\) are correctly derived from the original expression \((3y - 6)^2\).

Key Concepts

Binomial Theorem

The binomial theorem provides us with a powerful way to expand expressions of the form \((a + b)^n\). The theorem uses a formula that simplifies the process of expanding binomials by dealing with the powers of both terms in the binomial. When approached step-by-step, this allows us to find the expanded form of any binomial raised to a power without directly multiplying out large expressions.

For example, in the exercise \((3y - 6)^2\), we're using a special case of the binomial theorem tailored for squaring a binomial term. It's a great introduction to working with binomials because it focuses specifically on the pattern that emerges when \(n = 2\).

• The core idea is to leverage the predictable pattern in the expansion to simplify our calculations.
• Understanding this theorem better equips us to approach more complex polynomial expressions.

By recognizing and utilizing the theorem's structured approach, you can master the expansion of binomials efficiently.

Squaring a Binomial

Squaring a binomial involves expanding an expression of the form \((a - b)^2\), which is a foundational skill in algebra. Doing this allows us to convert the squared binomial into a sum of terms, utilizing the formula \(a^2 - 2ab + b^2\).

This formula results from multiplying the binomial by itself, thus allowing us to break it down into smaller, more manageable pieces. In the given exercise \((3y - 6)^2\), you're essentially applying this blueprint to find out what it expands to.

• Each part of the formula has a specific purpose:
  • \(a^2\): multiplies the first term by itself.
  • -2ab: represents twice the product of both terms.
  • \(b^2\): squares the second term.
• By splitting the binomial, you gain insight into how each part contributes to the expansion.

Recognizing this pattern helps not only in simplifying expressions but also in understanding the inherent relationship between algebraic terms.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the building blocks of algebra, making it possible to describe general patterns and solve equations.

Working with algebraic expressions involves manipulating these elements to simplify or transform them into a desired format, like expanding binomials or factoring polynomials.

• Understanding the way these expressions are constructed is crucial to grasp many algebraic concepts.
• Expressions like \((3y - 6)^2\) introduce a structured method to approach more complex algebraic problems.

By practicing the manipulation of algebraic expressions, such as from our example where we expanded into \(9y^2 - 36y + 36\), you're not only simplifying them but also enhancing your ability to solve a wide range of mathematical problems. These skills are integral in mathematics, allowing us to interpret and solve real-world issues efficiently.