Preliminaries: Real Numbers, Exponents, Factoring, and Analytic Geometry

Real Number System and Properties

• Real numbers comprise all numbers positioned on the real number line, encompassing both rational and irrational numbers.

• Classifications of Real Numbers:

  • Natural Numbers: The set of positive counting numbers starting from $1$:     $\{1, 2, 3, 4, \dots\}$
  • Integers: The set containing whole numbers, zero, and negative whole numbers:     $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$
  • Rational Numbers: Numbers that can be written as a ratio of two integers in the form $\frac{a}{b}$ where $b \neq 0$.
  • Examples: $\frac{1}{2}$, $-\frac{7}{3}$
  • Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers. Their decimal representations are non-terminating and non-repeating.
  • Examples: $\sqrt{2}$, $\pi$

• Properties of Real Numbers:

  • Commutative Property of Addition: The order of addition does not affect the sum:     $a + b = b + a$
  • Commutative Property of Multiplication: The order of multiplication does not affect the product:     $a \cdot b = b \cdot a$
  • Associative Property of Addition: The grouping of terms does not affect the sum:     $(a + b) + c = a + (b + c)$
  • Associative Property of Multiplication: The grouping of factors does not affect the product:     $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
  • Distributive Property: Multiplication distributes over addition:     $a \cdot (b + c) = a \cdot b + a \cdot c$

Finite and Infinite Intervals

• An interval represents a continuous subset of real numbers located on the number line.

• Finite Intervals:

  • Open Interval $(a, b)$: Defined as $\{x \mid a < x < b\}$. Endpoints $a$ and $b$ are excluded.
  • Example: $(-2, 1)$
  • Closed Interval $[a, b]$: Defined as $\{x \mid a \le x \le b\}$. Endpoints $a$ and $b$ are included.
  • Example: $[-1, 2]$
  • Half-Open Interval $(a, b]$: Defined as $\{x \mid a < x \le b\}$. Endpoint $a$ is excluded; endpoint $b$ is included.
  • Example: $\left(\frac{1}{2}, 3\right]$
  • Half-Open Interval $[a, b)$: Defined as $\{x \mid a \le x < b\}$. Endpoint $a$ is included; endpoint $b$ is excluded.
  • Example: $\left[-\frac{1}{2}, 3\right)$

• Infinite Intervals:
  • $(a, \infty)$: Defined as $\{x \mid x > a\}$.
  • Example: $(2, \infty)$
  • $[a, \infty)$: Defined as $\{x \mid x \ge a\}$.
  • Example: $[-1, \infty)$
  • $(-\infty, a)$: Defined as $\{x \mid x < a\}$.
  • Example: $(-\infty, 1)$
  • $(-\infty, a]$: Defined as $\{x \mid x \le a\}$.
  • Example: $\left(-\infty, -\frac{1}{2}\right]$

Rules and Laws of Exponents

• Definitions for $a^n$ (where base $a > 0$):

  • Integer Exponent: If $n$ is a positive integer, then $a^n$ represents $n$ factors of $a$ multiplied together:     $a^n = a \cdot a \cdot a \cdot \dots \cdot a$
  • Example: $2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$
  • Zero Exponent: Any non-zero base raised to the power of zero equals $1$:     $a^0 = 1$
  • Example: $7^0 = 1$
  • Exception: $0^0$ is not defined.
  • Negative Exponent: If $n$ is a positive integer and $a \neq 0$:     $a^{-n} = \frac{1}{a^n}$
  • Example: $6^{-2} = \frac{1}{6^2} = \frac{1}{36}$
  • Fractional Exponents:
  • a. For a positive integer $n$, $a^{1/n}$ or $\sqrt[n]{a}$ denotes the $n\text{th}$ root of $a$.
    • Example: $16^{1/2} = \sqrt{16} = 4$
  • b. For positive integers $m$ and $n$:       $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$
    • Example: $8^{2/3} = (\sqrt[3]{8})^2 = (2)^2 = 4$
  • c. For positive integers $m$ and $n$ (with $a \neq 0$):       $a^{-m/n} = \frac{1}{a^{m/n}}$
    • Example: $9^{-3/2} = \frac{1}{9^{3/2}} = \frac{1}{(\sqrt{9})^3} = \frac{1}{3^3} = \frac{1}{27}$

• Laws of Exponents:

  1. Product Rule: $a^m \cdot a^n = a^{m+n}$
  • Example: $x^2 \cdot x^3 = x^{2+3} = x^5$
  1. Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$ (where $a \neq 0$)
  • Example: $\frac{x^7}{x^4} = x^{7-4} = x^3$
  1. Power of a Power Rule: $(a^m)^n = a^{m \cdot n}$
  • Example: $(x^4)^3 = x^{4 \cdot 3} = x^{12}$
  1. Power of a Product Rule: $(ab)^n = a^n \cdot b^n$
  • Example: $(2x)^4 = 2^4 \cdot x^4 = 16x^4$
  1. Power of a Quotient Rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ (where $b \neq 0$)
  • Example: $\left(\frac{x}{2}\right)^3 = \frac{x^3}{2^3} = \frac{x^3}{8}$

• Common Pitfalls:

  • Avoid misapplying the product rule to powers of powers: $(x^2)^3 \neq x^5$.
  • The correct expression multiplies the exponents: $(x^2)^3 = x^{2 \cdot 3} = x^6$

Algebraic Expressions and Polynomials

• Algebraic Expressions in Calculus:

  • Expressions commonly encountered include:     $2x^{4/3} - x^{1/3} + 1$$2x^2 - x - \frac{2}{\sqrt{x}}$$\frac{3xy + 2}{x + 1}$$2x^3 + 2x + 1$

• Monomials and Polynomials:

  • Monomial: An algebraic expression of the form $ax^n$, where the coefficient $a$ is a real number and $n$ is a nonnegative integer. A monomial contains exactly one term.
  • Example: $7x^2$
  • Polynomial: A monomial or the sum of two or more monomials.
  • Examples of polynomials:       $x^2 + 4x + 4$$x^3 + 5$$x^4 + 3x^2 + 3$$x^2 y + xy + y$

• Like (Similar) Terms:

  • Constant terms and terms that share the exact same variable factors are classified as like terms.
  • Like terms can be combined by adding or subtracting their numerical coefficients.
  • Examples:     $3x + 7x = 10x$$\frac{1}{2}xy + 3xy = \frac{7}{2}xy$
  • This operation is mathematically justified by the distributive property of real numbers: $ab + ac = a(b + c)$.

Factoring Algebraic Expressions

• Factoring is the procedure of writing an algebraic expression as a product of simpler algebraic expressions.

• Factoring Out the Greatest Common Term (GCF):

  • Always inspect an algebraic expression to check for common factors across all terms before applying other techniques.
  • Example 1: $3x^2 - x = x(3x - 1)$
  • Example 2: Factor $2a^2 x + 4ax + 6a$. The common factor across all terms is $2a$:     $2a^2 x + 4ax + 6a = 2a \cdot ax + 2a \cdot 2x + 2a \cdot 3 = 2a(ax + 2x + 3)$

• Factoring Second-Degree Polynomials ($px^2 + qx + r$):

  • Second-degree polynomials with integral coefficients factor into $(ax + b)(cx + d)$, provided:     $ac = p$$ad + bc = q$$bd = r$
  • A trial-and-error method is systematically used to find factors.

• Detailed Factoring Procedure for $x^2 - 2x - 3$:

  • Step 1: Because the coefficient of $x^2$ is $1$, the first-degree terms in each factor must be $x$:     $(x \quad)(x \quad)$
  • Step 2: Identify integer pairs whose product equals the constant term $-3$. The possible pairs are $(-1, 3)$ and $(1, -3)$, yielding candidate factor sets:     $(x - 1)(x + 3)$$(x + 1)(x - 3)$
  • Step 3: Evaluate the inner and outer products to determine which set yields the linear term coefficient of $-2$:
  • Testing $(x - 1)(x + 3)$: Sum of outer and inner product coefficients is $(-1)(1) + (1)(3) = -1 + 3 = 2$
  • Testing $(x + 1)(x - 3)$: Sum of outer and inner product coefficients is $(1)(1) + (1)(-3) = 1 - 3 = -2$
  • Conclusion: The correct factorization is:     $x^2 - 2x - 3 = (x + 1)(x - 3)$

Quadratic Equations and the Quadratic Formula

• A quadratic equation written in standard form is given by:   $ax^2 + bx + c = 0$   where $a \neq 0$.

• The Quadratic Formula:

  • The explicit solutions for any quadratic equation in standard form are calculated using:     $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Cartesian Coordinate System and Lines

• Cartesian Coordinate System:
  • Coordinate geometry (analytic geometry) represents geometric figures using ordered numerical pairs.
  • Formed by two perpendicular number lines intersecting at origin $O(0, 0)$:
  • Horizontal axis: $x\text{-axis}$
  • Vertical axis: $y\text{-axis}$
  • Any point $P$ in two-dimensional space is designated as an ordered pair $P(x, y)$.

• Distance Formula:

  • The straight-line distance $d$ between two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ in the Cartesian plane is:     $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Slope of a Nonvertical Line:

  • The slope $m$ measures the steepness and direction of line $L$ passing through points $(x_1, y_1)$ and $(x_2, y_2)$, where $x_1 \neq x_2$:     $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

• Summary of Equations of Straight Lines:
  • Vertical Line: $x = a$
  • Horizontal Line: $y = b$
  • Point-Slope Form: $y - y_1 = m(x - x_1)$
  • Slope-Intercept Form: $y = mx + b$ (where $m$ is the slope and $(0, b)$ is the $y\text{-intercept}$)
  • General Form of a Linear Equation: $Ax + By + C = 0$ (where $A$, $B$, and $C$ are constants, and $A$ and $B$ are not both zero)

Worked Examples

• Example 0.1: Simplify the following exponent and radical expressions:

  • a. $(3x^2)(4x^3)$
  • Multiply coefficients and add exponents of matching bases:       $(3 \cdot 4)(x^{2+3}) = 12x^5$
  • b. $\frac{16^{5/4}}{16^{1/2}}$
  • Apply quotient law for exponents:       $16^{5/4 - 1/2} = 16^{5/4 - 2/4} = 16^{3/4}$
  • Simplify $16^{3/4}$:       $16^{3/4} = (\sqrt[4]{16})^3 = (2)^3 = 8$
  • c. $(6^{2/3})^3$
  • Apply power of a power law:       $6^{(2/3) \cdot 3} = 6^2 = 36$
  • d. $(x^3 y^{-2})^{-2}$
  • Distribute exponent across factors:       $x^{3(-2)} y^{-2(-2)} = x^{-6} y^4 = \frac{y^4}{x^6}$
  • e. $\left(\frac{y^{3/2}}{x^{1/4}}\right)^{-2}$
  • Apply negative power to numerator and denominator:       $\frac{y^{(3/2)(-2)}}{x^{(1/4)(-2)}} = \frac{y^{-3}}{x^{-1/2}} = \frac{x^{1/2}}{y^3} = \frac{\sqrt{x}}{y^3}$

• Example 0.2: Factorize the following algebraic expressions:

  • a. $2ax + 2ay + bx + by$
  • Group terms into pairs:       $(2ax + 2ay) + (bx + by)$
  • Factor out common terms from each group:       $2a(x + y) + b(x + y)$
  • Factor out binomial $(x + y)$:       $(2a + b)(x + y)$
  • b. $3x\sqrt{y} - 4 - 2\sqrt{y} + 6x$
  • Rearrange terms to group common factors:       $(3x\sqrt{y} + 6x) - (2\sqrt{y} + 4)$
  • Factor out common terms:       $3x(\sqrt{y} + 2) - 2(\sqrt{y} + 2)$
  • Factor out common binomial $(\sqrt{y} + 2)$:       $(3x - 2)(\sqrt{y} + 2)$

• Example 0.3: Factorize the following quadratic polynomials:

  • a. $3x^2 + 4x - 4$
  • Test factors of form $(3x + b)(x + d)$ where $b \cdot d = -4$:       $(3x - 2)(x + 2)$
  • Check middle term: $3x(2) + (-2)(x) = 6x - 2x = 4x$.
  • Correct factorization: $(3x - 2)(x + 2)$
  • b. $3x^2 - 6x - 24$
  • Factor out greatest numerical common factor $3$:       $3(x^2 - 2x - 8)$
  • Factor the remaining quadratic expression $x^2 - 2x - 8$ into two binomials:       $3(x - 4)(x + 2)$

• Example 0.4: Solve each quadratic equation:

  • a. $2x^2 + 5x - 12 = 0$
  • Identify coefficients: $a = 2$, $b = 5$, $c = -12$
  • Substitute into quadratic formula:       $x = \frac{-5 \pm \sqrt{5^2 - 4(2)(-12)}}{2(2)}$$x = \frac{-5 \pm \sqrt{25 + 96}}{4} = \frac{-5 \pm \sqrt{121}}{4} = \frac{-5 \pm 11}{4}$
  • Calculate solutions:       $x_1 = \frac{-5 + 11}{4} = \frac{6}{4} = \frac{3}{2}$$x_2 = \frac{-5 - 11}{4} = \frac{-16}{4} = -4$
  • b. $x^2 = -3x + 8$
  • Rearrange into standard form $ax^2 + bx + c = 0$:       $x^2 + 3x - 8 = 0$
  • Identify coefficients: $a = 1$, $b = 3$, $c = -8$
  • Substitute into quadratic formula:       $x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-8)}}{2(1)}$$x = \frac{-3 \pm \sqrt{9 + 32}}{2} = \frac{-3 \pm \sqrt{41}}{2}$

• Example 0.5 (Applied Distance and Cost Problem):

• Problem: Power relay station $S$ is positioned on a straight coastal highway at coordinates $S(10000, 0)$ (in feet). A marine biology experimental station $M$ is located on an island at coordinates $M(0, 3000)$ (in feet). A cable is laid from $S(10000, 0)$ along land to point $Q(2000, 0)$, and then under water from $Q(2000, 0)$ directly to $M(0, 3000)$. Land cable costs $\$3.00\text{ per running foot}$, while underwater cable costs $\$5.00\text{ per running foot}$. Calculate the total cost for laying the cable.

• Step 1: Determine the length of land cable from $S(10000, 0)$ to $Q(2000, 0)$:     $d_{\text{land}} = 10000 - 2000 = 8000\,\text{feet}$

• Step 2: Determine the length of underwater cable from $Q(2000, 0)$ to $M(0, 3000)$ using the distance formula:     $d_{\text{water}} = \sqrt{(0 - 2000)^2 + (3000 - 0)^2} = \sqrt{(-2000)^2 + (3000)^2}$$d_{\text{water}} = \sqrt{4,000,000 + 9,000,000} = \sqrt{13,000,000} = 1000\sqrt{13}\,\text{feet} \approx 3605.551\,\text{feet}$

• Step 3: Calculate total cable installation cost:     $\text{Cost}_{\text{land}} = 8000 \cdot 3.00 = \$24,000.00$$\text{Cost}_{\text{water}} = 1000\sqrt{13} \cdot 5.00 = 5000\sqrt{13} \approx \$18,027.76$$\text{Total Cost} = 24,000 + 18,027.76 = \$42,027.76$

  • Example 0.6: Find an equation of the line passing through point $(1, 3)$ with slope $m = 2$

• Use point-slope form $y - y_1 = m(x - x_1)$:     $y - 3 = 2(x - 1)$$y - 3 = 2x - 2$$y = 2x + 1$

  • Example 0.7: Find an equation of the line passing through points $(-3, 2)$ and $(4, -1)$

• Step 1: Calculate slope $m$:     $m = \frac{-1 - 2}{4 - (-3)} = \frac{-3}{7} = -\frac{3}{7}$

• Step 2: Apply point-slope form using point $(-3, 2)$:     $y - 2 = -\frac{3}{7}(x - (-3))$$y - 2 = -\frac{3}{7}(x + 3)$$y - 2 = -\frac{3}{7}x - \frac{9}{7}$$y = -\frac{3}{7}x + \frac{5}{7}$(In general form: $3x + 7y - 5 = 0$)

  • Example 0.8 (Applied Linear Sales Prediction):

• Problem: A sales manager plotted annual store sales versus time for the past 5 years and noted that the points lie approximately along a straight line. Using data points corresponding to Year 1 ($20\text{ thousand dollars}$) and Year 5 ($60\text{ thousand dollars}$), establish the equation of the trend line and predict sales for Year 6.
• Step 1: Identify coordinate points: $(1, 20)$ and $(5, 60)$.
• Step 2: Compute slope $m$:     $m = \frac{60 - 20}{5 - 1} = \frac{40}{4} = 10\,\text{thousand dollars per year}$
• Step 3: Determine line equation using point $(1, 20)$:     $y - 20 = 10(x - 1)$$y - 20 = 10x - 10$$y = 10x + 10$
• Step 4: Predict sales for Year 6 ($x = 6$):     $y = 10(6) + 10 = 60 + 10 = 70\,\text{thousand dollars}$
• The predicted sales figure for the sixth year is $\$70,000$.