Problem 54 - Algebra A Combined Function

Question

Factor each completely. $$ 9 y^{2}+12 y+4-x^{2} $$

Step-by-Step Solution

Answer

The expression factors to \((3y + 2 - x)(3y + 2 + x)\).

Step 1: Recognize the Structure

Take a look at the expression given: \(9y^{2} + 12y + 4 - x^{2}\). Notice that the expression can be seen as a difference between the trinomial \(9y^{2} + 12y + 4\) and the term \(x^2\). This suggests we might attempt to rewrite it in a form that involves the difference of squares in the subsequent steps.

Step 2: Factor the Quadratic Trinomial

First, observe the trinomial part \(9y^{2} + 12y + 4\) within the expression. This is a perfect square trinomial. It can be rewritten as \((3y + 2)^{2}\) because \((3y + 2)(3y + 2) = 9y^{2} + 12y + 4\).

Step 3: Substitute Back into the Expression

Now, substitute back the squared term into the original expression: \((3y + 2)^{2} - x^{2}\). Now the expression is in the form of a difference of squares, which is \((a^2 - b^2)\).

Step 4: Apply the Difference of Squares Formula

Utilize the difference of squares formula, which is \(a^2 - b^2 = (a - b)(a + b)\). In our case, \(a = 3y + 2\) and \(b = x\). Substituting these values gives us the factorization: \((3y + 2 - x)(3y + 2 + x)\).

Key Concepts

Difference of Squares

The difference of squares is a fundamental concept in algebra used to factor certain types of expressions. Consider two perfect squares, positioned such that one is subtracted from the other. This setup is what defines a difference of squares:

• A general example is expressed as \(a^2 - b^2\).
• This expression is the difference (subtraction) between \(a^2\) (a squared) and \(b^2\) (b squared).

The technique becomes useful because it allows us to factorize these expressions easily. When you identify a difference of squares, you can use a specific formula to simplify them, often turning a complex polynomial into something much more manageable. Keep an eye out for expressions like \((3y + 2)^2 - x^2\), where one term can be viewed as a perfect square being subtracted from another perfect square. Such instances are prime opportunities to apply the difference of squares formula.

Perfect Square Trinomial

A perfect square trinomial is a polynomial that can be represented as the square of a binomial. This simplifies our work, particularly when factoring, because:

• The trinomial typically takes the form \(a^2 + 2ab + b^2\).
• It can be rewritten as \((a + b)^2\).
• Similarly, \(a^2 - 2ab + b^2\) can be rewritten as \((a - b)^2\).
• This means the trinomial is compactly factored into a binomial squared.

In the exercise, the trinomial \((9y^2 + 12y + 4)\) is a perfect square because it matches the format above. Specifically, you can express it as \((3y + 2)^2\), saving time and effort in further calculations. Recognizing and rewriting perfect square trinomials helps in simplifying and solving algebraic expressions efficiently.

Quadratic Expression

A quadratic expression is a polynomial of degree 2, usually in the form \(ax^2 + bx + c\). It encompasses several potential forms that allow for different factoring techniques such as difference of squares and perfect square trinomials.

• The standard quadratic involves three terms, each with varying powers of the variable, typically capped at a square (second degree).
• Recognizing quadratics is vital because they serve as the backbone in algebra for a wide range of applications, from graphing parabolic curves to solving equations analytically.

In the exercise at hand, even though the primary expression contains another term (\(- x^2\)), the quadratic portion \((9y^2 + 12y + 4)\) remains central to its manipulation and transformation. This clear segregation of the quadratic portion allows us to apply specific factoring techniques like identifying a perfect square trinomial.

Difference of Squares Formula

The difference of squares formula is a powerful tool in factoring. This algebraic identity states that: \[a^2 - b^2 = (a - b)(a + b)\]. This allows any expression that fits the difference of squares format to be factored swiftly and neatly into a product of two binomials.

• For example, if you have \(9y^2 + 12y + 4 - x^2\), after recognizing it involves perfect squares, use the formula directly.
• In our case, \(a = (3y + 2)\) and \(b = x\), simplifying to \((3y + 2 - x)(3y + 2 + x)\).
• This reduction eases the calculation, especially in complex expressions where direct expansion would be tedious.

Using the difference of squares formula not only tidies up expressions but also highlights algebraic structures within more complicated polynomials, becoming an extensive aid in solving and simplifying polynomial equations.