Question

Factor: \(16 a^{2}-9 b^{2}\) (A) \((2 a+b)(a-b)\) (B) \(\quad(3 a+2 b)(6 a-3 b)\) (C) \(\quad(4 a+3 b)(4 a-3 b)\) (D) \(\quad(8 a+3 b)(2 a-3 b)\)

Short answer

The expression factors to \((4a + 3b)(4a - 3b)\), which is option (C).

Step-by-step solution

Recognize the Formula

The expression provided is a difference of squares, which generally takes the form \(x^2 - y^2\). The difference of squares can be factored using the formula: \(x^2 - y^2 = (x+y)(x-y)\).

Identify the Squares

Identify the perfect squares in the expression. Here, \(16a^2\) is a perfect square, which can be written as \((4a)^2\), and \(9b^2\) is also a perfect square, which can be written as \((3b)^2\).

Apply the Difference of Squares Formula

Using the formula \(x^2 - y^2 = (x+y)(x-y)\), substitute \(4a\) for \(x\) and \(3b\) for \(y\). The factored form is \((4a + 3b)(4a - 3b)\).

Compare with Given Options

Now, compare the factored form \((4a + 3b)(4a - 3b)\) with the options provided in the question. The option that matches this result is (C).

Key concepts

Understanding the Difference of Squares

One of the core concepts in algebraic factoring is the difference of squares. This technique applies when you have an expression in the form of \(x^2 - y^2\). The key idea here is that these two terms are both perfect squares, and they are being subtracted from each other, thus forming a 'difference'.

The beautiful and straightforward formula for the difference of squares is:

• \(x^2 - y^2 = (x+y)(x-y)\)

This means that you can express the difference as a product of two binomials, \((x + y)\) and \((x - y)\). This factoring technique is very useful because it transforms a single expression into a multiplication of two simpler expressions. Remember, this only works when you truly have a difference, not a sum, of two perfect squares.

Exploring Perfect Squares

Perfect squares are numbers or expressions that are the square of an integer or variable expression. For example, \(4, 9,\) and \(16\) are perfect squares of the numbers \(2, 3,\) and \(4\) respectively because \(2^2=4\), \(3^2=9\), and \(4^2=16\). Similarly, in algebra, expressions like \(x^2\) or \(4a^2\) are considered perfect squares because they can be expressed as \((x)^2\) or \((2a)^2\).

Spotting perfect squares is essential for using the difference of squares technique effectively. In the exercise, the terms \(16a^2\) and \(9b^2\) are perfect squares, written as \((4a)^2\) and \((3b)^2\). Recognizing these allows you to apply the difference of squares formula accurately. Perfect squares also appear in other factoring techniques, such as factoring trinomials, where they can help simplify complex expressions.

Mastering Factoring Techniques

Factoring techniques are strategies used to break down expressions into products of simpler expressions. One reason we factor is to simplify problems, especially when solving equations. There are several factoring techniques, and understanding these can make algebra much easier.

Some common factoring techniques include:

• **Common Factor Factoring:** This involves factoring out the greatest common factor from all terms in an expression.
• **Difference of Squares:** Already discussed, this technique deals with expressions that show a difference between two perfect squares.
• **Factoring Perfect Square Trinomials:** This involves expressions that are the square of a binomial, such as \(a^2 + 2ab + b^2\) which factors into \((a+b)^2\).
• **Factoring by Grouping:** Used when an expression has four terms that can be grouped into pairs, each pair containing a common factor.

Incorporating these techniques into your math toolkit equips you with powerful tools for simplifying and solving polynomial equations. Each technique can be recognized via specific patterns in expressions, opening the door to faster, more intuitive solutions.