Problem 17 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$9 a^{2}-(2 b+3)^{2}$$

Step-by-Step Solution

Answer

The factored expression is \((3a + 2b + 3)(3a - 2b - 3)\).

Step 1: Identify the Difference of Squares Pattern

We need to factor the expression using the difference of squares pattern, which is given by the formula \(A^2 - B^2 = (A + B)(A - B)\). Thus, we must rewrite the given expression in the form \(A^2 - B^2\).

Step 2: Express the Terms as Squares

The expression is \(9a^2 - (2b + 3)^2\). Notice that \(9a^2 = (3a)^2\) and the term \((2b + 3)^2\) is already a square. Thus, we recognize it as \((3a)^2 - (2b + 3)^2\).

Step 3: Apply the Difference of Squares Formula

Now that the expression is in the form \((3a)^2 - (2b + 3)^2\), apply the difference of squares formula: \((A + B)(A - B)\) where \(A = 3a\) and \(B = (2b + 3)\). This gives us: \[(3a + (2b + 3))(3a - (2b + 3))\]

Step 4: Simplify the Expression

Further simplify the factored expression to remove excess parentheses: \[(3a + 2b + 3)(3a - 2b - 3)\] Now the expression is fully factored.

Key Concepts

Factoring in Algebra

Factoring is a method used in algebra to rewrite an expression as a product of its factors. When you factor an expression, you break it down into its simplest components to make it easier to work with. This process doesn't change the value of the expression but offers a simpler form.
In algebra, there are specific patterns and identities, such as the "difference of squares," which help identify and factor polynomials efficiently.

• The difference of squares pattern is one of the fundamental identities: it notes how any expression of the form \(A^2 - B^2\) can be factored into \((A + B)(A - B)\).
• This identity is particularly useful because it transforms a difference into a multiplication operation, simplifying the expression.

Recognizing these patterns helps in reducing complex computations in algebraic expressions, making it a critical skill to master in algebra.

Polynomial Expressions

Polynomial expressions are sums of terms consisting of variables raised to whole number powers, often combined with coefficients. These expressions are fundamental in algebra and form the basis of many mathematical solutions.

• A simple example is the expression \(x^2 + 5x + 6\), where each term has the variable \(x\) raised to a certain power.
• Polynomials like \(9a^2 - (2b + 3)^2\) often require techniques such as factoring to simplify or solve equations that include them.

Sometimes, a polynomial expression appears in a form that matches a known identity or special factoring rule. Recognizing these allows you to immediately factor the expression without extensive calculations. The process of factoring polynomials often involves identifying the most efficient approach or pattern to apply.

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables involved. They are essential tools in algebra that simplify calculations and make problem-solving more intuitive.

• Main examples include the difference of squares, sum and difference of cubes, and perfect square trinomials.
• These identities allow students to break down and simplify complex algebraic expressions using recognized patterns.

Using these identities, complex expressions can be reduced to simpler forms or products. In our exercise, recognizing the expression \(9a^2 - (2b + 3)^2\) as a difference of squares allowed us to quickly apply the identity \(A^2 - B^2 = (A + B)(A - B)\), simplifying the work significantly and minimizing potential errors.