Precalculus Fundamentals: Algebraic Expressions, Exponents, Radicals, Polynomials, and Rational Expressions

P.1 Algebraic Expressions, Mathematical Models, and Real Numbers

Evaluate Algebraic Expressions

Evaluating algebraic expressions involves substituting given values for variables and simplifying the result using the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

• Order of operations: Always follow PEMDAS.

• Left to right: Operations of the same rank are performed left to right.

Use Mathematical Models

Mathematical models are formulas that describe real-world situations and relationships between variables.

Find the Intersection and Union of Sets

• Intersection: The set of elements common to both sets. Denoted as $A \cap B$.

• Union: The set of all elements in either set. Denoted as $A \cup B$.

Example: $A = \{3,4,6,7\}$, $B = \{3,6,7,9\}$ $A \cap B = \{3,6,7\}$ $A \cup B = \{3,4,6,7,9\}$

Recognize Subsets of the Real Numbers

The real numbers include several important subsets:

Subset	Definition	Examples
Natural Numbers ($\mathbb{N}$)	Counting numbers	1, 2, 3, ...
Whole Numbers	Natural numbers and zero	0, 1, 2, 3, ...
Integers ($\mathbb{Z}$)	Whole numbers and their negatives	..., -2, -1, 0, 1, 2, ...
Rational Numbers ($\mathbb{Q}$)	Numbers expressible as $\frac{a}{b}$, $b \neq 0$	$\frac{1}{2}$, -3, 0.75
Irrational Numbers	Cannot be written as $\frac{a}{b}$	$\sqrt{2}$, $\pi$
Real Numbers ($\mathbb{R}$)	All rational and irrational numbers	All above

Use Inequality Symbols

• <: Less than

• >: Greater than

• ≤: Less than or equal to

• ≥: Greater than or equal to

Evaluate Absolute Value

The absolute value $|a|$ is the distance from $a$ to 0 on the number line.

• If $a \geq 0$, $|a| = a$

• If $a < 0$, $|a| = -a$

Example: $|7 - 2| = 5$

Use Absolute Value to Express Distance

The distance between two points $a$ and $b$ on a number line is $|a - b|$.

Identify Properties of the Real Numbers

These properties apply to algebraic expressions:

Name	Meaning	Example
Commutative (Addition)	$a + b = b + a$	$7 + 4 = 4 + 7$
Commutative (Multiplication)	$ab = ba$	$3 \times 5 = 5 \times 3$
Associative (Addition)	$(a + b) + c = a + (b + c)$	$(2 + 3) + 4 = 2 + (3 + 4)$
Associative (Multiplication)	$(ab)c = a(bc)$	$(2 \times 3) \times 4 = 2 \times (3 \times 4)$
Distributive	$a(b + c) = ab + ac$	$2(3 + 4) = 2 \times 3 + 2 \times 4$
Identity (Addition)	$a + 0 = a$	$5 + 0 = 5$
Identity (Multiplication)	$a \times 1 = a$	$7 \times 1 = 7$

Simplify Algebraic Expressions

• Combine like terms and use distributive property as needed.

• Follow the order of operations.

Example: $6 + 2x - (x - 2) = 6 + 2x - x + 2 = 8 + x$

P.2 Exponents and Scientific Notation

Use the Product Rule

When multiplying exponential expressions with the same base, add the exponents:

$a^m \cdot a^n = a^{m+n}$

Use the Quotient Rule

When dividing exponential expressions with the same base, subtract the exponents:

$\frac{a^m}{a^n} = a^{m-n}$

Use the Zero-Exponent Rule

Any nonzero number raised to the zero power is 1:

$a^0 = 1$ (for $a \neq 0$)

Use the Negative-Exponent Rule

A negative exponent indicates the reciprocal:

$a^{-n} = \frac{1}{a^n}$ (for $a \neq 0$)

Use the Power Rule

When raising an exponential expression to a power, multiply the exponents:

$(a^m)^n = a^{mn}$

Find the Power of a Product

Raise each factor to the power:

$(ab)^n = a^n b^n$

Find the Power of a Quotient

Raise both numerator and denominator to the power:

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Simplify Exponential Expressions

• Remove parentheses using the power rule.

• Combine like bases using product and quotient rules.

• Express answers with positive exponents only.

Use Scientific Notation

Scientific notation expresses very large or small numbers as $a \times 10^n$, where $1 \leq |a| < 10$ and $n$ is an integer.

• Positive $n$: decimal moves right.

• Negative $n$: decimal moves left.

Example: $160,000 = 1.6 \times 10^5$; $0.0000008 = 8 \times 10^{-7}$

P.3 Radicals and Rational Exponents

Evaluate Square Roots

• The radical sign $\sqrt{\ }$ denotes the principal (nonnegative) square root.

• $\sqrt{a^2} = |a|$ for any real number $a$.

Simplify Expressions of the Form $\sqrt{a^2}$

The principal square root of $a^2$ is $|a|$.

Use the Product Rule to Simplify Square Roots

$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ for $a, b \geq 0$.

Use the Quotient Rule to Simplify Square Roots

$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ for $a \geq 0, b > 0$.

Add and Subtract Square Roots

• Combine like radicals (same radicand and index).

• Example: $\sqrt{72} - 3\sqrt{8} + 9\sqrt{18} = 3\sqrt{2} - 6\sqrt{2} + 27\sqrt{2} = 24\sqrt{2}$

Rationalize Denominators

• Multiply numerator and denominator by a value that eliminates radicals in the denominator.

• Example: $\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

Evaluate and Perform Operations with Higher Roots

• The $n$th root of $a$ is written $\sqrt[n]{a}$, where $n$ is a positive integer.

• If $n$ is even, $\sqrt[n]{a}$ is defined for $a \geq 0$.

• If $n$ is odd, $\sqrt[n]{a}$ is defined for all real $a$.

Understand and Use Rational Exponents

• $a^{1/n} = \sqrt[n]{a}$

• $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$

Example: $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$

P.4 Polynomials

Understanding the Vocabulary of Polynomials

• Monomial: One term (e.g., $5x$)

• Binomial: Two terms (e.g., $x^2 + 3x$)

• Trinomial: Three terms (e.g., $x^2 + 3x + 2$)

• Degree: The highest exponent of the variable in the polynomial

Adding and Subtracting Polynomials

• Combine like terms (same variable and exponent).

• Example: $(x^2 + 4x + 5) + (3x^2 - 2x - 15) = 4x^2 + 2x - 10$

Multiplying Polynomials

• Use distributive property or FOIL (First, Outside, Inside, Last) for binomials.

• Example: $(x + 2)(x + 4) = x^2 + 6x + 8$

Special Products in Polynomial Multiplication

• Square of a sum: $(a + b)^2 = a^2 + 2ab + b^2$

• Square of a difference: $(a - b)^2 = a^2 - 2ab + b^2$

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

Operations with Polynomials in Several Variables

• The degree is the sum of the exponents of all variables in a term.

• The degree of the polynomial is the highest degree among its terms.

P.5 Factoring Polynomials

Factor out the Greatest Common Factor (GCF)

• The GCF is the largest expression that divides each term of the polynomial.

• Example: $5x^2 - 15x = 5x(x - 3)$

Factor by Grouping

• Group terms with common factors and factor each group.

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

Factor Trinomials

• Find two numbers whose product is $ac$ and whose sum is $b$ in $ax^2 + bx + c$.

• Split the middle term and factor by grouping.

Factor the Difference of Squares

Use the identity $a^2 - b^2 = (a + b)(a - b)$.

Factor Perfect Square Trinomials

Use the identities $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$.

Factor the Sum or Difference of Cubes

• Sum: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

• Difference: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

General Strategy for Factoring Polynomials

1. Factor out the GCF.

2. Look for special products (difference of squares, perfect square trinomials, sum/difference of cubes).

3. Factor trinomials.

4. Check if further factoring is possible.

P.6 Rational Expressions

Specify Numbers that Must be Excluded from the Domain

A rational expression is undefined when its denominator is zero. Exclude values that make the denominator zero.

Example: $\frac{7x}{3x - 14}$ is undefined when $3x - 14 = 0 \implies x = \frac{14}{3}$

Simplify Rational Expressions

1. Factor numerators and denominators completely.

2. Divide out common factors.

Multiply Rational Expressions

1. Factor all numerators and denominators.

2. Multiply numerators together and denominators together.

3. Simplify by canceling common factors.

Divide Rational Expressions

1. Rewrite division as multiplication by the reciprocal.

2. Follow multiplication steps.

Add and Subtract Rational Expressions

1. Find the least common denominator (LCD).

2. Rewrite each expression with the LCD as the denominator.

3. Add or subtract numerators.

4. Simplify the result.

Simplify Complex Rational Expressions

• Method 1: Combine into a single fraction and divide.

• Method 2: Multiply numerator and denominator by the LCD of all rational expressions involved.