The analysis of rational expressions requires a rigorous application of algebraic factoring and domain restriction. We are tasked with simplifying the expression ##\frac{x^2 – 9}{x^2 + 5x + 6}## and identifying the specific values of ##x## that must be excluded from the domain to ensure the function remains well-defined within the real number system. This problem evaluates the interaction between polynomial decomposition and the foundational rules of division, particularly the prohibition of division by zero.

Algebraic Decomposition of Rational Functions

Factoring the Numerator via Difference of Squares

The numerator of the given rational expression is the binomial ##x^2 – 9##. This expression adheres to the structural form of a difference of squares, which is defined by the algebraic identity ###a^2 – b^2 = (a – b)(a + b)###. In this specific case, the variable term ##x^2## represents ##a^2##, implying ##a = x##, while the constant term ##9## represents ##b^2##, implying ##b = 3##. Recognizing this pattern is essential for reducing the complexity of the polynomial before attempting any cancellation of terms within the quotient.

By applying the difference of squares formula directly to the numerator, we obtain the factored form ###x^2 – 9 = (x – 3)(x + 3)###. This decomposition reveals the two primary linear factors that constitute the quadratic numerator. Each factor represents a potential root of the numerator polynomial; specifically, the numerator equals zero when ##x## is either ##3## or ##-3##. Understanding these roots is the first step in mapping the behavior of the rational function across the horizontal axis and identifying potential shared factors with the denominator.

Mathematically, we verify this factorization through the distributive property, often referred to as the FOIL method. Multiplying the first terms gives ##x \cdot x = x^2##, the outer terms gives ##x \cdot 3 = 3x##, the inner terms gives ##-3 \cdot x = -3x##, and the last terms gives ##-3 \cdot 3 = -9##. Summing these products results in ##x^2 + 3x – 3x – 9##, which simplifies back to ##x^2 – 9##. This confirms the validity of the factors ##(x – 3)## and ##(x + 3)## as the building blocks of our numerator expression.

Factoring the Trinomial Denominator

The denominator of the expression is the quadratic trinomial ##x^2 + 5x + 6##. To factor a monic trinomial of the form ##ax^2 + bx + c## where ##a = 1##, we seek two integers whose product equals the constant term ##c## and whose sum equals the linear coefficient ##b##. In this instance, we require two numbers that multiply to ##6## and add to ##5##. Analyzing the factor pairs of ##6##—namely ##(1, 6)## and ##(2, 3)##—we find that the pair ##(2, 3)## satisfies both the multiplicative and additive requirements of the coefficients.

Using these identified integers, we can rewrite the denominator in its factored form as ###x^2 + 5x + 6 = (x + 2)(x + 3)###. This transformation is critical because it highlights the specific linear components that dictate the vertical behavior of the rational function. The factors ##(x + 2)## and ##(x + 3)## indicate that the denominator will evaluate to zero at ##x = -2## and ##x = -3##. These specific values are the focal points for determining the domain constraints and the existence of any singularities or discontinuities within the function’s graph.

The accuracy of this factorization can be proven by expanding the binomials through algebraic multiplication. Applying the distributive law yields ##x(x + 3) + 2(x + 3)##, which expands further to ##x^2 + 3x + 2x + 6##. Combining the like terms ##3x## and ##2x## results in the original trinomial ##x^2 + 5x + 6##. This verification step ensures that no computational errors were made during the decomposition phase and provides a solid foundation for the subsequent simplification and identification of excluded values within the rational framework.

Identification and Analysis of Domain Constraints

Determining Excluded Values from the Denominator

In the study of rational expressions, the domain consists of all real numbers except those that cause the denominator to equal zero. Division by zero is undefined in mathematics because there is no real number that, when multiplied by zero, yields a non-zero numerator. To identify these excluded values, we must examine the original denominator before any simplification occurs. We set the quadratic expression ##x^2 + 5x + 6## equal to zero and solve for the variable ##x## using the factored form derived in the previous section.

By applying the Zero Product Property to the equation ###(x + 2)(x + 3) = 0###, we conclude that the denominator vanishes when either ##x + 2 = 0## or ##x + 3 = 0##. Solving these linear equations gives the solutions ##x = -2## and ##x = -3##. Consequently, these two values are strictly excluded from the domain of the rational expression. In mathematical notation, the domain can be expressed as ##\{x \in \mathbb{R} \mid x \neq -2, x \neq -3\}##, ensuring that the function remains defined for all other real inputs.

It is vital to identify these excluded values early in the simplification process. If we were to simplify the expression first and then determine the domain, we might overlook the value that was canceled out. For example, a computer program processing this function might encounter a runtime error if these values are not handled correctly. In a programming environment, an uninitialized or undefined state might be represented as ##””## or a null pointer, but in algebra, we must explicitly state the numerical constraints that prevent the expression from becoming logically inconsistent.

Removable Discontinuities vs. Infinite Discontinuities

The two excluded values, ##x = -2## and ##x = -3##, represent different types of discontinuities in the function. A removable discontinuity, or a hole, occurs when a factor appears in both the numerator and the denominator, allowing it to be canceled. In our expression, the factor ##(x + 3)## is present in both polynomials. This means that as ##x## approaches ##-3##, the expression approaches a specific finite value, yet the function itself remains technically undefined at that exact point because the original denominator would still be zero.

Conversely, an infinite discontinuity, or a vertical asymptote, occurs when a factor remains only in the denominator after the expression has been simplified. The factor ##(x + 2)## does not have a corresponding pair in the numerator, meaning it cannot be removed through algebraic cancellation. As ##x## approaches ##-2##, the denominator approaches zero while the numerator remains a non-zero constant. This causes the absolute value of the expression to grow without bound, manifesting as a vertical line on the coordinate plane that the graph never actually touches.

The distinction between these two singularities is significant for the topology of the function. The hole at ##x = -3## is a point where the curve is interrupted but maintains a consistent limit from both sides. The asymptote at ##x = -2## represents a fundamental break in the continuity where the function’s output diverges toward positive or negative infinity. By classifying these excluded values, we gain a comprehensive understanding of the function’s behavior near its limits and can accurately predict the visual characteristics of its graphical representation in a Cartesian system.

Simplification and Formal Expression of the Quotient

Cancellation of Common Factors and Equivalence

With the numerator and denominator fully factored, we can now proceed to the simplification phase. The original expression is written as ###\frac{(x – 3)(x + 3)}{(x + 2)(x + 3)}###. Algebraically, any term divided by itself is equal to one, provided that the term is not zero. Since we have already established that ##x \neq -3##, the factor ##(x + 3)## in the numerator and the factor ##(x + 3)## in the denominator can be divided out. This process reduces the complexity of the expression while preserving its mathematical meaning for all values in the domain.

The resulting simplified expression is ###\frac{x – 3}{x + 2}###. It is crucial to note that while this simplified version looks like a standard rational function, it is only equivalent to the original expression under the condition that ##x \neq -3##. Without this caveat, the two expressions are not identical because the original function is undefined at ##x = -3##, whereas the simplified version would evaluate to ##(-3 – 3) / (-3 + 2) = -6 / -1 = 6##. This discrepancy highlights the importance of keeping track of the original domain constraints.

In formal mathematical writing, the final answer should be stated clearly with the simplified form and the associated exclusions. The final result of the simplification is ###\frac{x – 3}{x + 2}, \quad x \neq -2, -3###. This presentation provides the most efficient version of the function while simultaneously warning the reader of the specific points where the function fails to exist. This rigorous approach prevents errors in subsequent calculus operations, such as differentiation or integration, where the presence of a hole or an asymptote would require specialized treatment or limit evaluation.

Graphing the Rational Function and Point Holes

Visualizing the simplified expression ##y = \frac{x – 3}{x + 2}## helps to clarify the relationship between the algebraic factors and the geometric curve. The graph of this function is a hyperbola with a vertical asymptote at ##x = -2## and a horizontal asymptote at ##y = 1##. The horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and denominator, which in this case is ##1/1##. The ##x##-intercept occurs at ##x = 3##, where the numerator is zero, and the ##y##-intercept occurs at ##y = -1.5##, found by evaluating the function at ##x = 0##.

To accurately represent the original expression, a hole must be placed on the graph at the coordinate point associated with ##x = -3##. By plugging ##x = -3## into the simplified expression ##\frac{x – 3}{x + 2}##, we find the ##y##-coordinate of the hole: ###\frac{-3 – 3}{-3 + 2} = \frac{-6}{-1} = 6###. Therefore, the graph of the original expression is the hyperbola ##y = \frac{x – 3}{x + 2}## with a visible open circle at the point ##(-3, 6)##. This open circle signifies that the point is excluded from the function’s set of outputs.

The behavior of the function near the vertical asymptote at ##x = -2## also requires careful plotting. As ##x## approaches ##-2## from the right, the numerator ##x-3## is approximately ##-5## and the denominator ##x+2## is a very small positive number, leading to a limit of negative infinity. As ##x## approaches ##-2## from the left, the denominator is a very small negative number, causing the quotient to approach positive infinity. These directional limits provide a complete picture of the function’s trajectory and emphasize the non-removable nature of the infinite discontinuity at that location.

Theoretical Deep Dive and Mathematical Context

Algebraic Properties of Rational Fields

Rational expressions are elements of what is known in abstract algebra as a field of fractions. Given a ring of polynomials ##\mathbb{R}[x]##, we can construct the field of rational functions ##\mathbb{R}(x)## by considering all possible quotients of these polynomials. This structure is analogous to how the field of rational numbers ##\mathbb{Q}## is constructed from the ring of integers ##\mathbb{Z}##. In this context, simplifying a rational expression is equivalent to finding a representative of an equivalence class in the simplest possible form, where two fractions are considered equivalent if their cross-products are equal.

The process of factoring used in this problem relies on the unique factorization property of the polynomial ring ##\mathbb{R}[x]##. Because every polynomial over a field can be uniquely factored into irreducible components (up to a constant factor), we are guaranteed that our decomposition of the numerator and denominator is the only possible way to break down the expression into linear terms. This uniqueness is a cornerstone of algebraic theory, allowing mathematicians to solve complex equations by reducing them to simpler, linear factors that are much easier to manage and interpret within various computational frameworks.

Furthermore, the study of rational functions extends into the realm of complex analysis. If we allowed ##x## to be a complex number, the excluded values ##x = -2## and ##x = -3## would be referred to as poles or singularities. The hole at ##x = -3## would be considered a removable singularity because the limit exists and is finite. The asymptote at ##x = -2## is a simple pole. This deeper perspective allows for the application of Cauchy’s Residue Theorem and other advanced techniques to evaluate integrals that would otherwise be impossible to solve using standard real-number calculus alone.

Limits and Continuity in Rational Functions

From the perspective of real analysis, the continuity of a rational function is a local property that holds at every point where the denominator is non-zero. A function ##f(x)## is continuous at a point ##c## if the limit as ##x## approaches ##c## equals the function’s value ##f(c)##. In our problem, the function is continuous on the intervals ##(-\infty, -3)##, ##(-3, -2)##, and ##(-2, \infty)##. At the excluded values, the function fails the definition of continuity because it is not defined at those points, regardless of whether a limit exists from the left and right.

The limit evaluation at the removable discontinuity is a classic exercise in introductory calculus. We write ###\lim_{x \to -3} \frac{x^2 – 9}{x^2 + 5x + 6} = \lim_{x \to -3} \frac{x – 3}{x + 2}###. Substituting ##-3## into the simplified form yields the value ##6##. This limit exists because the factor causing the zero-over-zero indeterminacy has been removed. However, even though the limit exists, the function remains discontinuous at ##x = -3## because the value of the function is not defined there. This distinction is subtle but critical for understanding the formal definition of mathematical continuity.

In summary, the simplification of the expression ##\frac{x^2 – 9}{x^2 + 5x + 6}## into ##\frac{x – 3}{x + 2}## with exclusions at ##x = -2, -3## serves as an essential bridge between basic algebra and advanced analysis. By meticulously factoring the polynomials and identifying the domain constraints, we transform a complex quotient into a manageable function while respecting the fundamental laws of arithmetic. This technical precision ensures that any further mathematical operations performed on the expression are both accurate and logically sound, reflecting the high standards of technical STEM architecture and algebraic rigor.