Question 1

Factoring the Numerator via Difference of Squares

Accepted Answer

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.

Question 2

Factoring the Trinomial Denominator

Accepted Answer

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.

Question 3

Determining Excluded Values from the Denominator

Accepted Answer

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.

Question 4

Removable Discontinuities vs. Infinite Discontinuities

Accepted Answer

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.

Question 5

Cancellation of Common Factors and Equivalence

Accepted Answer

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 
eq -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.

Question 6

Graphing the Rational Function and Point Holes

Accepted Answer

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##.

Question 7

Algebraic Properties of Rational Fields

Accepted Answer

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.

Question 8

Limits and Continuity in Rational Functions

Accepted Answer

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.

JUPITER SCIENCE

Simplify ## \frac{x^2 – 9}{x^2 + 5x + 6} ##

Constructing Continuous Functions: Examples and Proofs

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