Precalculus Study Guide: Numbers, Expressions, Equations, and Functions

Numbers and Number Sets

Types of Numbers

Understanding the different types of numbers is foundational in precalculus. Numbers are classified into several sets, each with unique properties.

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

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

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

• Rational Numbers: Numbers that can be expressed as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

• Irrational Numbers: Numbers that cannot be expressed as a fraction, such as $\sqrt{2}$ or $\pi$.

• Real Numbers: All rational and irrational numbers.

Example: $\frac{3}{4}$ is rational; $\sqrt{5}$ is irrational.

Prime Factorization and Least Common Multiple (LCM)

Prime Factorization

Prime factorization is expressing a number as a product of its prime factors.

• Prime Number: A number greater than 1 with only two positive divisors: 1 and itself.

• Example: The prime factorization of 156 is $2^2 \times 3 \times 13$.

Least Common Multiple (LCM)

The LCM of two or more numbers is the smallest number that is a multiple of each.

• Example: LCM of 8, 24, and 40 is 120.

Algebraic Expressions and Operations

Evaluating and Simplifying Expressions

Algebraic expressions can be simplified using properties of arithmetic and exponents.

• Order of Operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS).

• Example: $3y^2$ is a simplified form of $3y \times y$.

Factoring Polynomials

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

• Common Methods: Factoring out the greatest common factor (GCF), grouping, difference of squares, trinomials.

• Example: $x^2 - 6x + 9 = (x - 3)^2$

Scientific Notation

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10.

• Example: $0.000102 = 1.02 \times 10^{-4}$

Radicals and Rationalizing the Denominator

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots.

• Example: $\sqrt{27y^4} = 3y^2$

Rationalizing the Denominator

Rationalizing the denominator means rewriting a fraction so that the denominator contains no radicals.

• Method: Multiply numerator and denominator by a suitable radical to eliminate the radical in the denominator.

• Example: $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Equations and Inequalities

Solving Linear Equations

Linear equations are equations of the first degree, typically in the form $ax + b = 0$.

• Example: Solve $x^2 = 4x$ by factoring: $x(x - 4) = 0 \implies x = 0$ or $x = 4$

Solving for a Variable

Isolating a variable in an equation involves algebraic manipulation.

• Example: Solve for $F$ in $C = F$

Solving Word Problems

Word problems require translating real-world scenarios into mathematical equations.

• Example: Two cars travel in opposite directions. If one travels 5 mph faster than the other and they are 267 miles apart after 3 hours, set up equations to solve for their speeds.

Functions and Graphs

Graphing Linear Equations

Linear equations can be graphed on the coordinate plane. The general form is $y = mx + b$.

• Slope ($m$): Measures the steepness of the line.

• Y-intercept ($b$): The point where the line crosses the y-axis.

• Example: Graph $2x + y = 8$ by solving for $y$ and plotting points.

Finding the Equation of a Line

Given two points, the equation of the line can be found using the point-slope form.

• Formula: $y - y_1 = m(x - x_1)$, where $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: Find the equation of the line through $(1, -3)$ and $(2, 5)$.

Properties of Exponents and Radicals

Exponent Rules

Exponents follow specific rules for multiplication, division, and powers.

• 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: $a^{-n} = \frac{1}{a^n}$

Radical and Exponential Notation

Radicals can be expressed in exponential form: $\sqrt[n]{a} = a^{1/n}$

• Example: $\sqrt[3]{x^4} = x^{4/3}$

Factoring and Simplifying Complex Fractions

Factoring Techniques

Factoring is used to simplify expressions and solve equations.

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

• Trinomials: $ax^2 + bx + c$ can often be factored into $(x + p)(x + q)$

Complex Fractions

Complex fractions have fractions in the numerator, denominator, or both. Simplify by finding a common denominator.

• Example: $\frac{\frac{2}{x} - 1}{\frac{1}{x} - 2}$

Summary Table: Key Properties and Operations

Operation	Key Property	Example
Addition/Subtraction	Combine like terms	$3x + 2x = 5x$
Multiplication	Distributive property	$a(b + c) = ab + ac$
Division	Divide coefficients and subtract exponents	$\frac{x^5}{x^2} = x^{3}$
Factoring	Find GCF, use patterns	$x^2 - 9 = (x - 3)(x + 3)$
Radicals	Express as exponents	$\sqrt{x} = x^{1/2}$
Scientific Notation	Express as $a \times 10^n$	$0.0005 = 5 \times 10^{-4}$

Additional info:

• Some questions involve graphing and interpreting linear equations, which is a core skill in precalculus.

• Problems on rationalizing denominators and simplifying radicals reinforce understanding of exponents and roots.

• Factoring polynomials and solving equations are essential for progressing to higher-level algebra and calculus.