Problem 95 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The factorized form is \((x + 2 + 3y)(x + 2 - 3y)\).

Step 1: Recognize the Expression Structure

The polynomial given is \(x^2 + 4x + 4 - 9y^2\). Notice it resembles the form \((a+b)^2 - c^2\) which is a difference of squares: \( (a+b)^2 - (c)^2 = (a+b+c)(a+b-c) \).

Step 2: Factor the Quadratic

First, let's focus on the expression \(x^2 + 4x + 4\). This is a perfect square trinomial, and can be factored as \((x + 2)^2\) because \((x+2)(x+2) = x^2 + 4x + 4\).

Step 3: Identify the Expression as a Difference of Squares

Now, the expression is \((x + 2)^2 - 9y^2\). Recognize this as a difference of squares, where \((x + 2)^2\) is the first square and \(9y^2\) is the second square, with \(c = 3y\) because \((3y)^2 = 9y^2\).

Step 4: Apply the Difference of Squares Formula

Apply the difference of squares formula: \((a + b)^2 - (c)^2 = (a + b + c)(a + b - c)\). Here, \(a + b = x + 2\) and \(c = 3y\). So, it factors to: \((x + 2 + 3y)(x + 2 - 3y)\).

Key Concepts

Difference of Squares

The difference of squares is a handy factoring technique used to simplify polynomial expressions that are made up of two perfect squares separated by a subtraction sign. Suppose you have an expression in the form of \(a^2 - b^2\). You can factor it into \((a + b)(a - b)\).
This factoring method takes advantage of the idea that when two squares are involved, their difference can be split into a product of a sum and a difference.

• Example: \(x^2 - 9 = (x + 3)(x - 3)\).
• In our exercise, \( (x + 2)^2 - 9y^2\) can be rewritten as a difference of squares where \((x+2)\) and \(3y\) are squared terms.

Recognizing patterns like these helps in quickly simplifying expressions to their factored form, which is useful for solving equations further down the line.

Quadratic Trinomials

Quadratic trinomials are algebraic expressions that can usually be recognized by their standard form: \(ax^2 + bx + c\). A key skill in algebra is being able to factor these trinomials back into their binomial form.
For the given quadratic trinomial \(x^2 + 4x + 4\), noticing that it is a perfect square trinomial is essential. A perfect square trinomial is one that can be expressed as the square of a binomial:

• Example: \(x^2 + 4x + 4\) is equivalent to \((x + 2)^2\).

Factoring quadratic trinomials often helps in solving equations or simplifying expressions. In practice, you'll often use trial and error or formulas to identify the factors.

Factoring Techniques

Factoring techniques are essential tools in algebra, offering a way to break down complex expressions into simpler, more manageable parts. These techniques involve identifying common patterns or structures within an expression, allowing you to express them as products of simpler factors.
Several common factoring techniques include:

• Factoring out the Greatest Common Factor (GCF): Simplifying by dividing each term by a common factor.
• Differentiating quadratic trinomials: Such as recognizing perfect squares like in \((x+2)^2\).
• Applying difference of squares: Especially when expressions resemble \(a^2 - b^2\).

These techniques not only simplify the task of working through algebra problems but also enhance your problem-solving strategies. With practice, identifying and applying the appropriate factoring technique becomes second nature.