Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomial. \(324 x^{2}-121\)

Step-by-Step Solution

Answer

The factored form is \((18x - 11)(18x + 11)\).

Step 1: Identify the type of polynomial

The given polynomial is \(324x^2 - 121\). Notice that this is a difference of squares, which can be expressed in the form \(a^2 - b^2\).

Step 2: Express as a difference of squares

Recognize that \(324x^2\) is a perfect square \((18x)^2\) and \(121\) is also a perfect square \((11)^2\). Thus, the polynomial is structured as \((18x)^2 - (11)^2\).

Step 3: Apply the difference of squares formula

The formula for factoring a difference of squares is \(a^2 - b^2 = (a-b)(a+b)\). Applying this to \((18x)^2 - (11)^2\), we get \((18x - 11)(18x + 11)\).

Step 4: Verify the factorization

To ensure this factorization is correct, expand \((18x - 11)(18x + 11)\) back to \((18x)^2 - (11)^2\), which simplifies to \(324x^2 - 121\), confirming our factorization.

Key Concepts

Difference of Squares

The difference of squares is a common method used in polynomial factorization. It's based on the concept that if you have two perfect squares subtracted from one another, it can be factored using a specific formula. In algebra, this is expressed as \(a^2 - b^2\), where \(a^2\) and \(b^2\) are perfect squares. This can always be rewritten as \((a-b)(a+b)\).

The idea can be applied to many algebraic expressions, especially those where you notice two perfect squares. For example, consider the polynomial \(324x^2 - 121\). By recognizing that \(324x^2\) equals \((18x)^2\) and \(121\) equals \(11^2\), you see a difference of squares.

• Identify square terms: \((18x)^2 - (11)^2\)
• Apply the formula: \((18x - 11)(18x + 11)\)

This simple technique is widely used because it's direct and simplifies expressions efficiently.

Perfect Squares

Perfect squares are numbers or expressions that can be obtained by squaring another number or expression. In algebra, it's essential to recognize perfect squares because they simplify the process of factoring.

In the polynomial \(324x^2 - 121\), recognizing \((18x)^2\) and \(11^2\) as perfect squares is the first step towards factorization. This kind of recognition allows us to apply the difference of squares method easily.

When factoring, always look for:

• Numbers that have clean square roots, like 121 (since \(11^2 = 121\)).
• Variable terms squared, such as \((18x)^2\).

Understanding perfect squares streamlines factoring and aids in breaking down complex expressions into simpler forms.

Factoring Techniques

Factoring techniques are methods used to simplify polynomials and other algebraic expressions by rewriting them as products of simpler factors. These techniques serve as foundational tools in algebra and can be applied across various types of problems.

Different Factoring techniques include:

• Difference of squares: Used when the expression is in the form \(a^2 - b^2\), like in the exercise \((18x)^2 - 11^2\).
• Greatest common factor (GCF): Factoring out the common factor in all terms of the polynomial.
• Perfect square trinomials: Identify trinomials that fit the pattern of \((a \, \pm \, b)^2\).
• General trinomials: Factor trinomials of the form \(ax^2 + bx + c\).

These techniques assist in unveiling the factors of polynomials, making them easier to solve or analyze. By selecting the correct method for the specific type of polynomial, you can effectively reduce and simplify complex expressions.