Problem 30 - Precalculus: Functions and Graphs

Question

Factor the polynomial. $$x^{2}-4 y^{2}-6 x+9$$

Step-by-Step Solution

Answer

The polynomial cannot be factored easily into a simpler polynomial form.

Step 1: Identify the Structure of the Polynomial

The polynomial given is a quadratic trinomial: \(x^2 - 4y^2 - 6x + 9\). Notice that it can potentially be expressed as a difference of squares or factorable via other methods.

Step 2: Rearrange the Polynomial

Reorganize terms to make it easier to observe potential factoring structures: \( x^2 - 6x - 4y^2 + 9\).

Step 3: Consider Separating into Two Groups

Examine the expression and try grouping differently to facilitate factoring techniques. Let’s consider: \((x^2 - 6x) - (4y^2 - 9)\).

Step 4: Factor Each Group Separately

For the first group, \(x^2 - 6x\), factor out \(x\): \(x(x - 6)\). The second group, \(4y^2 - 9\), can be expressed as a difference of squares: \((2y)^2 - 3^2\). This factors to \((2y - 3)(2y + 3)\).

Step 5: Assemble the Factored Form

Putting it together, our polynomial remains split into two groups due to different variable parts. Combine the factored parts: \((x(x-6)) - ((2y-3)(2y+3))\). Thus, the polynomial cannot be traditionally factored any further due to this form.

Key Concepts

Quadratic Trinomial

A quadratic trinomial is a polynomial expression that consists of three terms, usually arranged in a standard form \( ax^2 + bx + c \). In this context, the polynomial \( x^2 - 6x - 4y^2 + 9 \) resembles a quadratic relationship with respect to the variable \( x \) but includes an additional term \( -4y^2 \), associated with a different variable \( y \).
This specific polynomial's complexity arises from combining different variable terms. When dealing with quadratic trinomials, always begin by identifying the single variable structure before exploring other factoring possibilities. In our example, focusing on \( x^2 - 6x \) as one part and \(-4y^2 + 9 \) as another helps to sort out possible methods for factoring.

Difference of Squares

The difference of squares is a special factoring technique applicable when an expression is of the form \( a^2 - b^2 \), which can be factored into \( (a-b)(a+b) \). In the polynomial \( 4y^2 - 9 \), notice that it fits the difference of squares pattern.
The term \( 4y^2 \) can be seen as \( (2y)^2 \) and \( 9 \) as \( 3^2 \). This allows us to factor it into \((2y - 3)(2y + 3)\).
Recognizing patterns such as these is key in simplifying complex polynomial expressions. Remember: spotting these differences can greatly enhance problem-solving efficiency, as it reduces complex expressions to simpler binomial factors.

Grouping Method

The grouping method is a valuable technique in factoring polynomials, especially when dealing with four terms or more. This method involves rearranging and partitioning terms into groups that are more manageable. For the expression \( x^2 - 6x - 4y^2 + 9 \), grouping is essential.
Begin by separating the polynomial into strategic groups that can individually be simplified further. Here, we consider the parts \( (x^2 - 6x) \) and \( (4y^2 - 9) \). This initial split suggests distinct factoring methods for each part – one using simple factoring
for a quadratic and the other using the difference of squares.
The goal of grouping is to expose further arithmetic relationships that reveal the polynomial’s true structure, simplifying the factoring process.

Polynomial Structure

Understanding the structure of a polynomial is crucial for determining suitable methods for simplification. The polynomial \( x^2 - 4y^2 - 6x + 9 \), upon rearrangement, reveals its compound nature – a blend of both quadratic and special products. This polynomial showcases a mix of variables \( x \) and \( y \), adding complexity to its analysis.
Recognizing these structures allows for selecting appropriate factoring strategies, such as viewing parts of it as quadratic or as a difference of squares.
Always take a closer look at the arrangement of the terms.

• Does it fit known factoring patterns like difference of squares?
• Is there a common factor adjustable for each term?

A thorough evaluation helps to manage and simplify the expressions effectively, paving the way for finding hidden relationships within the polynomial.