Precalculus Foundations: Numbers, Operations, Exponents, Radicals, and Polynomials

Number Sets and Classification

Types of Numbers

Understanding the classification of numbers is foundational in precalculus. Numbers are grouped into several sets based on their properties.

• Natural Numbers: Counting numbers starting from 1 (e.g., 1, 2, 3, ...).

• Whole Numbers: Natural numbers plus zero (e.g., 0, 1, 2, 3, ...).

• Integers: Whole numbers and their negatives (e.g., ..., -3, -2, -1, 0, 1, 2, 3, ...).

• Rational Numbers: Numbers that can be written as a fraction of two integers, where the denominator is not zero (e.g., 1/2, -3, 0.75).

• Irrational Numbers: Numbers that cannot be written as a fraction of two integers (e.g., $\sqrt{2}$, $\pi$).

• Real Numbers: All rational and irrational numbers.

Example: $-3.25$ is a rational number, $\sqrt{4}$ is a rational number (since $\sqrt{4} = 2$), and $\sqrt{6}$ is irrational.

Basic Arithmetic Operations

Addition, Subtraction, Multiplication, and Division

Mastery of arithmetic operations with integers, fractions, and decimals is essential for algebraic manipulation.

• Addition/Subtraction: Combine like terms or values. Pay attention to signs.

• Multiplication: Multiply numbers directly; for negatives, remember that a negative times a negative is positive.

• Division: Divide as usual; dividing by a negative changes the sign.

Example: $-11 + 13 = 2$; $-4 \times (-12) = 48$; $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$.

Order of Operations

PEMDAS/BODMAS

When evaluating expressions, follow the order of operations:

• Parentheses first

• Exponents (powers and roots, etc.)

• Multiplication and Division (left to right)

• Addition and Subtraction (left to right)

Example: $2 + 3 \times 4 = 2 + 12 = 14$

Properties of Numbers

Opposites, Reciprocals, and Absolute Value

• Opposite (Additive Inverse): The number that, when added to the original, yields zero. For $a$, the opposite is $-a$.

• Reciprocal (Multiplicative Inverse): The number that, when multiplied by the original, yields one. For $a \neq 0$, the reciprocal is $\frac{1}{a}$.

• Absolute Value: The distance from zero on the number line, always non-negative. $|a|$.

Example: The opposite of $-12$$is$12$; the reciprocal of$-12$is$-\frac{1}{12}$.

Exponents and Powers

Exponent Rules

Exponents are used to represent repeated multiplication. Key rules include:

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$

• Zero Exponent: $a^0 = 1$ (for $a \neq 0$)

Example: $x^{-3} = \frac{1}{x^3}$; $(x^2)^3 = x^6$.

Radicals and Roots

Simplifying Radicals

Radicals represent roots, such as square roots or cube roots. Simplifying involves factoring out perfect squares or cubes.

• Square Root: $\sqrt{a}$ is a number which, when squared, gives $a$.

• Product Rule for Radicals: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$

• Quotient Rule for Radicals: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

Example: $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

Polynomials

Operations with Polynomials

Polynomials are algebraic expressions with terms consisting of variables raised to whole number powers and their coefficients.

• Addition/Subtraction: Combine like terms.

• Multiplication: Use distributive property or FOIL for binomials.

• Special Products: $(a+b)^2 = a^2 + 2ab + b^2$; $(a-b)^2 = a^2 - 2ab + b^2$; $(a+b)(a-b) = a^2 - b^2$

Example: $(x+2)(x-3) = x^2 - x - 6$

Factoring Polynomials

Factoring is expressing a polynomial as a product of its factors.

• Greatest Common Factor (GCF): Factor out the largest common factor from all terms.

• Factoring Trinomials: $ax^2 + bx + c$ can often be factored into $(mx + n)(px + q)$.

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

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

Solving Equations

Linear and Quadratic Equations

Solving equations involves finding the value(s) of the variable that make the equation true.

• Linear Equations: $ax + b = 0$; solve by isolating $x$.

• Quadratic Equations: $ax^2 + bx + c = 0$; solve by factoring, completing the square, or using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Solve $x^2 - 5x + 6 = 0$ by factoring: $(x-2)(x-3) = 0 \implies x = 2, 3$

Fractions and Rational Expressions

Operations with Fractions

• Add/Subtract: Find a common denominator.

• Multiply: Multiply numerators and denominators.

• Divide: Multiply by the reciprocal of the divisor.

Example: $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$

Summary Table: Number Set Classification

Number	Natural	Whole	Integer	Rational	Irrational	Real
5	✔	✔	✔	✔		✔
0		✔	✔	✔		✔
-3			✔	✔		✔
$\frac{1}{2}$				✔		✔
$\sqrt{2}$					✔	✔

Additional info: This guide covers foundational topics from Ch. 1 (Graphs, Number Sets), Ch. 2 (Functions and Their Graphs, as related to polynomials), Ch. 3 (Linear and Quadratic Functions), and Ch. 4 (Polynomial and Rational Functions) of a typical Precalculus curriculum.