Review of Algebra: Fundamental Concepts for Precalculus

0. Fundamental Concepts of Algebra

0.1 Sets of Real Numbers

Understanding sets and the classification of real numbers is foundational in algebra and precalculus. A set is a collection of objects, and each object in a set is called an element of that set.

• Examples of Sets:

  • Set of positive integers: {1, 2, 3, ...}

  • Set of integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}

• Rational and Irrational Numbers:

  • The set of real numbers includes both rational and irrational numbers.

  • The real number line visually represents all real numbers, showing their order and relative position.

0.2 Some Properties of Real Numbers

Real numbers follow several important algebraic properties that are essential for manipulating expressions and solving equations.

• Transitive Property of Equality: If $a = b$ and $b = c$, then $a = c$.

• Closure Properties: For all real numbers $a$ and $b$, $a + b$ and $ab$ are also real numbers.

• Commutative Properties: $a + b = b + a$ and $ab = ba$.

• Associative Properties: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$.

• Identity Properties: $a + 0 = a$ and $a imes 1 = a$.

• Inverse Properties: For each $a$, there is $-a$ such that $a + (-a) = 0$; for $a \neq 0$, there is $a^{-1}$ such that $a a^{-1} = 1$.

• Distributive Property: $a(b + c) = ab + ac$.

Example: Applying the commutative and associative properties to rearrange and simplify expressions.

0.3 Exponents and Radicals

Exponents

Exponents represent repeated multiplication of a base. For a positive integer $n$:

• $x^n = x \times x \times \cdots \times x$ ($n$ factors)

• $x^0 = 1$

• $(x^m)^n = x^{mn}$

• $x^{-n} = \frac{1}{x^n}$ for $x \neq 0$

Examples:

• $3^5 = 243$

• $(-1)^4 = 1$

Radicals

A radical is an expression that involves roots. The $n$th root of $x$ is written as $\sqrt[n]{x}$.

• $\sqrt[n]{x^n} = x$

• $\sqrt{xy} = \sqrt{x} \sqrt{y}$

• $\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}$

Example: $\sqrt{25} = 5$; $\sqrt{(-3)^2} = 3$

0.4 Operations with Algebraic Expressions

Algebraic expressions are formed by combining numbers and variables using operations. A polynomial in $x$ is an expression of the form $c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0$.

• Example: $3x^2y - 2x + 1$ is an algebraic expression in $x$ and $y$.

Special Products

• $(x + y)^2 = x^2 + 2xy + y^2$ (square of a sum)

• $(x - y)^2 = x^2 - 2xy + y^2$ (square of a difference)

• $(x + y)(x - y) = x^2 - y^2$ (difference of squares)

Example: $(x + 2)(x - 5) = x^2 - 3x - 10$

0.5 Factoring

Factoring is the process of expressing an algebraic expression as a product of its factors.

• Common Factor: $xy + xz = x(y + z)$

• Perfect Square Trinomial: $x^2 + 2ax + a^2 = (x + a)^2$

• Difference of Squares: $x^2 - a^2 = (x + a)(x - a)$

• Sum/Difference of Cubes: $x^3 + a^3 = (x + a)(x^2 - ax + a^2)$

Example: Factor $3k^2x^2 + 9k^2x = 3k^2x(x + 3)$

0.6 Fractions

Simplifying Fractions

The fundamental principle of fractions allows us to multiply or divide the numerator and denominator by the same nonzero quantity.

• Fractions must have nonzero denominators.

• Multiplication: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

• Division: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

• Addition/Subtraction: Combine over a common denominator.

Example: $\frac{x^2 - x - 6}{x^2 + 7x + 12} = \frac{(x - 3)(x + 2)}{(x + 3)(x + 4)}$ for $x \neq -3$

0.7 Equations and Linear Equations

Equations

An equation is a statement that two expressions are equal, separated by an equality sign (=). A variable is a symbol that can be replaced by numbers.

• Equivalent Equations: Two equations with the same solutions.

• Operations that guarantee equivalence: Adding/subtracting the same polynomial, multiplying/dividing by a nonzero constant, replacing a side with an equal expression.

Linear Equations

A linear equation in $x$ can be written as $ax + b = 0$, where $a \neq 0$. It is also called a first-degree equation.

• Example: Solve $5x - 6 = 3x$ $5x - 6 = 3x$ $2x = 6$ $x = 3$

Literal, Fractional, and Radical Equations

• Literal Equations: Equations with constants represented by letters (e.g., $a, b, c$).

• Fractional Equations: Equations where the variable appears in the denominator.

• Radical Equations: Equations where the variable appears under a radical.

Example: Solve $\sqrt{y} - 3 = \sqrt{y} - 3$ for $y$.

0.8 Quadratic Equations

A quadratic equation in $x$ is of the form $ax^2 + bx + c = 0$, where $a \neq 0$. It is also called a second-degree equation.

• Factoring: $x^2 + x - 12 = 0$ factors as $(x - 3)(x + 4) = 0$ so $x = 3$ or $x = -4$.

• Quadratic Formula: The solutions to $ax^2 + bx + c = 0$ are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Quadratic-Form Equations: Equations that can be transformed into quadratic equations by substitution.

Example: Solve $4x - 4x^3 = 0$ $4x(1 - x^2) = 0$ $x = 0, 1, -1$