Chapter 1 — Prerequisites
Before diving into algebra and trigonometry, you need a solid foundation in the language of mathematics. This chapter reviews the fundamental building blocks: real numbers, exponents, polynomials, factoring, rational expressions, and radicals. Mastery here is essential — every future topic builds on these concepts.
Table of Contents
1 — Real Numbers: Algebra Essentials
- 1.1 The Number Line and Sets of Numbers
- 1.2 Order of Operations
- 1.3 Properties of Real Numbers
- 1.4 Evaluating Algebraic Expressions
- 1.5 Simplifying Algebraic Expressions
2 — Exponents and Scientific Notation
- 2.1 The Product Rule for Exponents
- 2.2 The Quotient Rule for Exponents
- 2.3 The Power Rule for Exponents
- 2.4 Zero Exponent Rule
- 2.5 Negative Exponents
- 2.6 Scientific Notation
3 — Radicals and Rational Exponents
- 3.1 Square Roots and Higher Roots
- 3.2 Rational Exponents
- 3.3 Simplifying Radical Expressions
- 3.4 Rationalizing the Denominator
4 — Polynomials
- 4.1 What Is a Polynomial?
- 4.2 Adding and Subtracting Polynomials
- 4.3 Multiplying Polynomials
- 4.4 Special Products
5 — Factoring Polynomials
- 5.1 Greatest Common Factor (GCF)
- 5.2 Factoring Trinomials
- 5.3 Factoring Special Products
- 5.4 Factoring by Grouping
- 5.5 General Strategy for Factoring
6 — Rational Expressions
Glossary — Key Terms at a Glance
| Term | Meaning | ||
|---|---|---|---|
| Natural Numbers | Counting numbers: $1, 2, 3, 4, \ldots$ | ||
| Whole Numbers | Natural numbers plus zero: $0, 1, 2, 3, \ldots$ | ||
| Integers | Whole numbers and negatives: $\ldots, -2, -1, 0, 1, 2, \ldots$ | ||
| Rational Numbers | Numbers expressible as $\frac{p}{q}$ where $p, q$ are integers and $q \neq 0$ | ||
| Irrational Numbers | Numbers that cannot be expressed as a ratio of integers (e.g., $\sqrt{2}, \pi$) | ||
| Real Numbers | The union of rational and irrational numbers | ||
| Exponent | A number indicating how many times a base is multiplied by itself | ||
| Polynomial | An expression of finite length made up of variables and coefficients using only addition, subtraction, multiplication, and non-negative integer exponents | ||
| Degree | The highest exponent of a variable in a polynomial | ||
| Factor | An expression that divides another expression evenly | ||
| Rational Expression | A fraction where numerator and/or denominator are polynomials | ||
| Radical | An expression involving a root (square root, cube root, etc.) | ||
| Radicand | The expression under the radical sign | ||
| Scientific Notation | A number written as $a \times 10^n$ where $1 \leq | a | < 10$ |
1 — Real Numbers: Algebra Essentials
1.1 The Number Line and Sets of Numbers
Numbers are the foundation of all mathematics. Understanding the different sets or families of numbers is critical.
Sets of Numbers (each is a subset of the next):
\[\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\]| Set | Symbol | Description | Examples |
|---|---|---|---|
| Natural Numbers | $\mathbb{N}$ | Counting numbers | $1, 2, 3, 4, \ldots$ |
| Whole Numbers | $\mathbb{W}$ | Natural numbers + zero | $0, 1, 2, 3, \ldots$ |
| Integers | $\mathbb{Z}$ | Whole numbers + negatives | $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ |
| Rational Numbers | $\mathbb{Q}$ | Ratios of integers | $\frac{1}{2}, -\frac{3}{4}, 0.75, 0.\overline{3}$ |
| Irrational Numbers | $\mathbb{R} \setminus \mathbb{Q}$ | Non-repeating, non-terminating decimals | $\sqrt{2}, \pi, e$ |
| Real Numbers | $\mathbb{R}$ | All rational + irrational numbers | Everything on the number line |
How to Classify a Number
Classify each number:
| Number | Classification |
|---|---|
| $-7$ | Integer, Rational, Real |
| $\frac{3}{5}$ | Rational, Real |
| $\sqrt{5}$ | Irrational, Real |
| $0$ | Whole, Integer, Rational, Real |
| $3.14$ | Rational (terminating decimal), Real |
| $0.\overline{6}$ | Rational (repeating decimal = $\frac{2}{3}$), Real |
| $\pi$ | Irrational, Real |
The Number Line
The number line is a visual representation of all real numbers:
- Every point on the line corresponds to exactly one real number
- Numbers increase from left to right
-
The distance between a number and zero is its absolute value: $ x $
Absolute Value:
\[|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}\]The absolute value of a number is always non-negative. It represents distance from zero.
\[|5| = 5, \quad |-5| = 5, \quad |0| = 0\]1.2 Order of Operations
When an expression involves more than one operation, we follow a strict order to ensure everyone gets the same answer.
PEMDAS (order of operations):
| Priority | Operation | Notes |
|---|---|---|
| 1st | Parentheses | Innermost first; includes brackets $[ ]$ and braces ${ }$ |
| 2nd | Exponents | Including roots |
| 3rd | Multiplication & Division | Left to right (equal priority) |
| 4th | Addition & Subtraction | Left to right (equal priority) |
⚠️ Common Mistake: Multiplication does NOT always come before division. They have equal priority, and you evaluate left to right. Same for addition and subtraction.
\[24 \div 6 \times 2 = 4 \times 2 = 8 \quad \text{(not } 24 \div 12 = 2\text{)}\]Worked Example:
Evaluate $5 + 3 \times 2^3 - (4 + 1)$:
| Step | Work | Result |
|---|---|---|
| 1. Parentheses | $5 + 3 \times 2^3 - \mathbf{5}$ | $(4+1) = 5$ |
| 2. Exponents | $5 + 3 \times \mathbf{8} - 5$ | $2^3 = 8$ |
| 3. Multiplication | $5 + \mathbf{24} - 5$ | $3 \times 8 = 24$ |
| 4. Add/Subtract (L→R) | $29 - 5 = \mathbf{24}$ | Final answer |
Nested Parentheses Example:
\[\frac{5^2 - 4}{7} \cdot \sqrt{3^2 + 4^2}\]| Step | Work |
|---|---|
| Numerator | $5^2 - 4 = 25 - 4 = 21$ |
| Denominator | $7$ |
| Fraction | $\frac{21}{7} = 3$ |
| Radical | $\sqrt{9 + 16} = \sqrt{25} = 5$ |
| Final | $3 \cdot 5 = \mathbf{15}$ |
1.3 Properties of Real Numbers
These properties are the axioms that govern how real numbers behave. They justify every algebraic manipulation you will ever perform.
| Property | Addition | Multiplication |
|---|---|---|
| Commutative | $a + b = b + a$ | $a \cdot b = b \cdot a$ |
| Associative | $(a + b) + c = a + (b + c)$ | $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ |
| Identity | $a + 0 = a$ | $a \cdot 1 = a$ |
| Inverse | $a + (-a) = 0$ | $a \cdot \frac{1}{a} = 1 \; (a \neq 0)$ |
| Distributive | \multicolumn{2}{c}{$a(b + c) = ab + ac$} |
Why These Matter: Every step in solving an equation is justified by one of these properties. When you “move a term to the other side,” you’re using the additive inverse property. When you “divide both sides,” you’re using the multiplicative inverse property.
Identify the property used:
| Statement | Property |
|---|---|
| $7 + 3 = 3 + 7$ | Commutative (Addition) |
| $4(x + 5) = 4x + 20$ | Distributive |
| $(2 \cdot 5) \cdot 3 = 2 \cdot (5 \cdot 3)$ | Associative (Multiplication) |
| $x + 0 = x$ | Identity (Addition) |
| $6 \cdot \frac{1}{6} = 1$ | Inverse (Multiplication) |
1.4 Evaluating Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and operations. To evaluate means to substitute given values for the variables and compute.
Variable: A letter or symbol representing an unknown number.
Algebraic Expression: A combination of variables, numbers, and at least one operation ($+, -, \times, \div$).
To evaluate an algebraic expression:
- Substitute the given value(s) for the variable(s)
- Simplify using the order of operations
Evaluate $3x^2 - 2x + 5$ when $x = 4$:
\[3(4)^2 - 2(4) + 5 = 3(16) - 8 + 5 = 48 - 8 + 5 = \mathbf{45}\]Evaluate $\frac{a^2 + b^2}{a - b}$ when $a = 5, \, b = 3$:
\[\frac{5^2 + 3^2}{5 - 3} = \frac{25 + 9}{2} = \frac{34}{2} = \mathbf{17}\]1.5 Simplifying Algebraic Expressions
Like Terms: Terms that have the same variable(s) raised to the same power(s).
- $3x^2$ and $-7x^2$ are like terms
- $3x^2$ and $3x$ are not like terms (different powers)
- $5xy$ and $-2xy$ are like terms
To simplify: Combine like terms by adding/subtracting their coefficients.
Simplify:
\[7x^2 - 3x + 2x^2 + 5x - 8\]Group like terms:
\[= (7x^2 + 2x^2) + (-3x + 5x) + (-8)\] \[= 9x^2 + 2x - 8\]Using the Distributive Property to Simplify:
\[3(2x - 5) + 4(x + 3)\] \[= 6x - 15 + 4x + 12\] \[= (6x + 4x) + (-15 + 12)\] \[= 10x - 3\]2 — Exponents and Scientific Notation
2.1 The Product Rule for Exponents
Exponential Notation: For any real number $a$ and natural number $n$:
\[a^n = \underbrace{a \cdot a \cdot a \cdots a}_{n \text{ factors}}\]Where:
- $a$ is the base
- $n$ is the exponent (or power)
Product Rule: When multiplying expressions with the same base, add the exponents:
\[a^m \cdot a^n = a^{m+n}\]Examples:
\[2^3 \cdot 2^5 = 2^{3+5} = 2^8 = 256\] \[x^4 \cdot x^7 = x^{4+7} = x^{11}\] \[3a^2b^3 \cdot 5a^4b = 15a^{2+4}b^{3+1} = 15a^6b^4\]⚠️ Common Mistake: The product rule only works with the same base!
\[2^3 \cdot 3^2 \neq 6^5\]You cannot combine $2^3 \cdot 3^2$ using the product rule. You must compute: $8 \cdot 9 = 72$.
2.2 The Quotient Rule for Exponents
Quotient Rule: When dividing expressions with the same base, subtract the exponents:
\[\frac{a^m}{a^n} = a^{m-n}, \quad a \neq 0\]2.3 The Power Rule for Exponents
Power Rule: When raising a power to another power, multiply the exponents:
\[(a^m)^n = a^{mn}\]Power of a Product:
\[(ab)^n = a^n b^n\]Power of a Quotient:
\[\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, \quad b \neq 0\]2.4 Zero Exponent Rule
Zero Exponent Rule: Any nonzero number raised to the power of zero equals 1:
\[a^0 = 1, \quad a \neq 0\]Why? Using the quotient rule: $\frac{a^n}{a^n} = a^{n-n} = a^0$. But also $\frac{a^n}{a^n} = 1$. Therefore $a^0 = 1$.
⚠️ $0^0$ is undefined. This is a convention agreed upon by mathematicians. Don’t confuse it with $a^0 = 1$ for $a \neq 0$.
But be careful with placement:
\[-5^0 = -(5^0) = -1 \quad \text{(exponent applies to 5 only)}\] \[(-5)^0 = 1 \quad \text{(exponent applies to } -5\text{)}\]2.5 Negative Exponents
Negative Exponent Rule: A negative exponent means “take the reciprocal”:
\[a^{-n} = \frac{1}{a^n}, \quad a \neq 0\] \[\frac{1}{a^{-n}} = a^n\]More generally:
\[\frac{a^{-m}}{b^{-n}} = \frac{b^n}{a^m}\]Examples:
\[3^{-2} = \frac{1}{3^2} = \frac{1}{9}\] \[x^{-5} = \frac{1}{x^5}\] \[\frac{a^{-3}}{b^{-2}} = \frac{b^2}{a^3}\] \[\frac{2x^{-3}y^4}{5x^2y^{-1}} = \frac{2y^4 \cdot y^1}{5x^2 \cdot x^3} = \frac{2y^5}{5x^5}\]Complete Exponent Rules Summary
| Rule | Formula |
|---|---|
| Product Rule | $a^m \cdot a^n = a^{m+n}$ |
| Quotient Rule | $\frac{a^m}{a^n} = a^{m-n}$ |
| Power Rule | $(a^m)^n = a^{mn}$ |
| Power of Product | $(ab)^n = a^n b^n$ |
| Power of Quotient | $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ |
| Zero Exponent | $a^0 = 1 \; (a \neq 0)$ |
| Negative Exponent | $a^{-n} = \frac{1}{a^n}$ |
🔍 Memorize this table. These seven rules are used constantly throughout algebra and trigonometry.
2.6 Scientific Notation
Scientific Notation: A number written as:
\[a \times 10^n\]| where $1 \leq | a | < 10$ and $n$ is an integer. |
- Positive $n$: the number is large (move decimal right)
- Negative $n$: the number is small (move decimal left)
| Standard Form | Scientific Notation | Direction |
|---|---|---|
| $4{,}560{,}000$ | $4.56 \times 10^6$ | Decimal moved 6 places left |
| $0.000073$ | $7.3 \times 10^{-5}$ | Decimal moved 5 places right |
| $327.8$ | $3.278 \times 10^2$ | Decimal moved 2 places left |
Operations with Scientific Notation:
Multiplication:
\[(3 \times 10^4)(5 \times 10^{-2}) = 15 \times 10^{4+(-2)} = 15 \times 10^2 = 1.5 \times 10^3\]Division:
\[\frac{8.4 \times 10^6}{2.1 \times 10^3} = 4 \times 10^{6-3} = 4 \times 10^3\]3 — Radicals and Rational Exponents
3.1 Square Roots and Higher Roots
Square Root: If $b^2 = a$, then $b = \sqrt{a}$.
\[\sqrt{a} \text{ is the principal (non-negative) square root of } a\]$n$th Root: The $n$th root of $a$ is written $\sqrt[n]{a}$. If $b^n = a$, then $b = \sqrt[n]{a}$.
| Root | Notation | Example |
|---|---|---|
| Square root | $\sqrt{a}$ or $a^{1/2}$ | $\sqrt{25} = 5$ |
| Cube root | $\sqrt[3]{a}$ or $a^{1/3}$ | $\sqrt[3]{27} = 3$ |
| Fourth root | $\sqrt[4]{a}$ or $a^{1/4}$ | $\sqrt[4]{81} = 3$ |
| $n$th root | $\sqrt[n]{a}$ or $a^{1/n}$ | General form |
Perfect squares to memorize:
| $n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $n^2$ | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 |
Perfect cubes to memorize:
| $n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| $n^3$ | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 |
⚠️ Key Facts:
-
$\sqrt{a^2} = a $ (not just $a$!) — the principal square root is always non-negative - $\sqrt[n]{a}$ for even $n$: $a$ must be $\geq 0$ (in real numbers)
- $\sqrt[n]{a}$ for odd $n$: $a$ can be any real number — $\sqrt[3]{-8} = -2$
- $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$ — this is a very common error!
3.2 Rational Exponents
Rational Exponents connect exponents and radicals:
\[a^{1/n} = \sqrt[n]{a}\] \[a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}\]You can think of the numerator as the power and the denominator as the root:
\[a^{m/n} = (\text{root first, then power}) \text{ or } (\text{power first, then root})\]Converting between radicals and rational exponents:
\[\sqrt[3]{x^5} = x^{5/3}\] \[8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4\] \[27^{-2/3} = \frac{1}{27^{2/3}} = \frac{1}{(\sqrt[3]{27})^2} = \frac{1}{3^2} = \frac{1}{9}\] \[16^{3/4} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8\]Strategy: When computing $a^{m/n}$, always take the root first, then the power. This keeps numbers small.
$64^{2/3} = (\sqrt[3]{64})^2 = 4^2 = 16$ is much easier than computing $64^2 = 4096$ first!
3.3 Simplifying Radical Expressions
Product Rule for Radicals:
\[\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b} \quad (a, b \geq 0 \text{ for even } n)\]Quotient Rule for Radicals:
\[\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \quad (b \neq 0)\]A radical expression is in simplest form when:
- No perfect $n$th power factors remain under the radical
- No fractions are under the radical
- No radicals appear in the denominator
Simplify $\sqrt{72}$:
\[\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}\]Simplify $\sqrt[3]{54x^5}$:
\[\sqrt[3]{54x^5} = \sqrt[3]{27 \cdot 2 \cdot x^3 \cdot x^2} = 3x\sqrt[3]{2x^2}\]Simplify $\sqrt{50a^3b^4}$:
\[\sqrt{50a^3b^4} = \sqrt{25 \cdot 2 \cdot a^2 \cdot a \cdot b^4} = 5ab^2\sqrt{2a}\]Adding and Subtracting Radicals
You can only combine like radicals (same index, same radicand):
3.4 Rationalizing the Denominator
Rationalizing means eliminating radicals from the denominator by multiplying by an appropriate form of 1.
For single-term denominators: Multiply by $\frac{\sqrt[n]{a^{n-k}}}{\sqrt[n]{a^{n-k}}}$.
For binomial denominators: Multiply by the conjugate.
\[\text{Conjugate of } (a + \sqrt{b}) \text{ is } (a - \sqrt{b})\]Single-term:
\[\frac{5}{\sqrt{3}} = \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}\]Binomial denominator:
\[\frac{4}{3 + \sqrt{2}} = \frac{4}{3 + \sqrt{2}} \cdot \frac{3 - \sqrt{2}}{3 - \sqrt{2}} = \frac{4(3 - \sqrt{2})}{9 - 2} = \frac{4(3 - \sqrt{2})}{7} = \frac{12 - 4\sqrt{2}}{7}\]Why does the conjugate work? Because of the difference of squares:
\[(a + b)(a - b) = a^2 - b^2\]When $b = \sqrt{c}$, we get $a^2 - c$, which eliminates the radical!
4 — Polynomials
4.1 What Is a Polynomial?
A polynomial in $x$ is an expression of the form:
\[a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0\]where:
- $a_n, a_{n-1}, \ldots, a_0$ are coefficients (real numbers)
- $n$ is a non-negative integer
- $a_n \neq 0$ ($a_n$ is the leading coefficient)
- $n$ is the degree of the polynomial
Classification by number of terms:
| Name | Terms | Example |
|---|---|---|
| Monomial | 1 term | $5x^3$ |
| Binomial | 2 terms | $x^2 - 4$ |
| Trinomial | 3 terms | $2x^2 + 3x - 7$ |
Classification by degree:
| Degree | Name | Example |
|---|---|---|
| 0 | Constant | $7$ |
| 1 | Linear | $3x + 2$ |
| 2 | Quadratic | $x^2 - 5x + 6$ |
| 3 | Cubic | $x^3 + 2x^2 - x$ |
| 4 | Quartic | $x^4 - 1$ |
| 5 | Quintic | $x^5 + x$ |
4.2 Adding and Subtracting Polynomials
To add or subtract polynomials: combine like terms (terms with the same variable and exponent).
Addition:
\[(3x^2 + 5x - 7) + (2x^2 - 3x + 4) = 5x^2 + 2x - 3\]Subtraction (distribute the negative!):
\[(4x^3 + 2x - 9) - (x^3 - 5x + 3)\] \[= 4x^3 + 2x - 9 - x^3 + 5x - 3\] \[= 3x^3 + 7x - 12\]4.3 Multiplying Polynomials
To multiply polynomials: Use the distributive property — every term in the first polynomial multiplies every term in the second.
For binomials, use the FOIL method: First, Outer, Inner, Last.
FOIL Example:
\[(2x + 3)(x - 5)\]| Step | Terms | Result |
|---|---|---|
| First | $2x \cdot x$ | $2x^2$ |
| Outer | $2x \cdot (-5)$ | $-10x$ |
| Inner | $3 \cdot x$ | $3x$ |
| Last | $3 \cdot (-5)$ | $-15$ |
Multiplying larger polynomials:
\[(x + 2)(x^2 - 3x + 4)\] \[= x \cdot x^2 + x \cdot (-3x) + x \cdot 4 + 2 \cdot x^2 + 2 \cdot (-3x) + 2 \cdot 4\] \[= x^3 - 3x^2 + 4x + 2x^2 - 6x + 8\] \[= x^3 - x^2 - 2x + 8\]4.4 Special Products
These patterns appear so frequently that you should memorize them:
| Pattern | Formula |
|---|---|
| Difference of Squares | $(a + b)(a - b) = a^2 - b^2$ |
| Perfect Square (Sum) | $(a + b)^2 = a^2 + 2ab + b^2$ |
| Perfect Square (Diff) | $(a - b)^2 = a^2 - 2ab + b^2$ |
| Cube of Sum | $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ |
| Cube of Diff | $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$ |
⚠️ Critical Mistakes to Avoid:
\[(a + b)^2 \neq a^2 + b^2 \quad \text{(missing the } 2ab \text{ term!)}\] \[(a - b)^2 \neq a^2 - b^2 \quad \text{(that's the difference of squares, not a perfect square)}\]Difference of Squares:
\[(3x + 5)(3x - 5) = (3x)^2 - 5^2 = 9x^2 - 25\]Perfect Square:
\[(4x - 3)^2 = (4x)^2 - 2(4x)(3) + 3^2 = 16x^2 - 24x + 9\]Cube of Sum:
\[(x + 2)^3 = x^3 + 3x^2(2) + 3x(4) + 8 = x^3 + 6x^2 + 12x + 8\]5 — Factoring Polynomials
Factoring is the reverse of multiplication — you’re breaking an expression into the product of simpler expressions. It’s one of the most important skills in algebra.
5.1 Greatest Common Factor (GCF)
Greatest Common Factor (GCF): The largest expression that divides all terms of the polynomial evenly.
Step 1: Find the GCF of all coefficients. Step 2: Find the lowest power of each common variable. Step 3: Factor it out.
Factor $6x^3 + 9x^2 - 15x$:
- GCF of $6, 9, 15$ is $3$
- Lowest power of $x$ is $x^1$
- GCF = $3x$
5.2 Factoring Trinomials
Factoring $x^2 + bx + c$: Find two numbers $p$ and $q$ such that:
- $p + q = b$ (they add to the middle coefficient)
- $p \cdot q = c$ (they multiply to the constant term)
Then: $x^2 + bx + c = (x + p)(x + q)$
Factoring $ax^2 + bx + c$ (when $a \neq 1$): Use the AC method:
- Multiply $a \cdot c$
- Find two numbers that multiply to $ac$ and add to $b$
- Rewrite the middle term and factor by grouping
Factor $x^2 + 7x + 12$:
Find two numbers: $p + q = 7$ and $p \cdot q = 12$
\[p = 3, \; q = 4 \quad \checkmark\] \[x^2 + 7x + 12 = (x + 3)(x + 4)\]Factor $2x^2 + 7x + 3$ (AC Method):
- $a \cdot c = 2 \cdot 3 = 6$
- Find two numbers: multiply to $6$, add to $7$ → $1$ and $6$
- Rewrite: $2x^2 + x + 6x + 3$
- Group: $(2x^2 + x) + (6x + 3) = x(2x + 1) + 3(2x + 1)$
- Factor: $(2x + 1)(x + 3)$
5.3 Factoring Special Products
| Pattern | Factored Form |
|---|---|
| Difference of Squares | $a^2 - b^2 = (a + b)(a - b)$ |
| Perfect Square Trinomial | $a^2 + 2ab + b^2 = (a + b)^2$ |
| Perfect Square Trinomial | $a^2 - 2ab + b^2 = (a - b)^2$ |
| Sum of Cubes | $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ |
| Difference of Cubes | $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ |
Mnemonic for cubes: SOAP
\[a^3 \pm b^3 = (a \; [\mathbf{S}ame] \; b)(a^2 \; [\mathbf{O}pposite] \; ab \; [\mathbf{A}lways \; \mathbf{P}ositive] \; b^2)\]- Same sign as the original
- Opposite sign
- Always Positive
Factor $27x^3 - 8$:
\[= (3x)^3 - 2^3 = (3x - 2)(9x^2 + 6x + 4)\]Factor $x^2 - 49$:
\[= x^2 - 7^2 = (x + 7)(x - 7)\]Factor $4x^2 + 12x + 9$:
\[= (2x)^2 + 2(2x)(3) + 3^2 = (2x + 3)^2\]⚠️ Sum of Squares: $a^2 + b^2$ is NOT factorable over the real numbers! Don’t try to factor it.
\[x^2 + 9 \neq (x + 3)(x - 3) \quad \text{(that gives } x^2 - 9\text{, not } x^2 + 9\text{)}\]5.4 Factoring by Grouping
Factoring by Grouping works for polynomials with 4 or more terms:
- Group terms in pairs
- Factor out the GCF from each pair
- If both groups share a common binomial factor, factor it out
Factor $x^3 + 3x^2 + 2x + 6$:
\[= (x^3 + 3x^2) + (2x + 6)\] \[= x^2(x + 3) + 2(x + 3)\] \[= (x + 3)(x^2 + 2)\]5.5 General Strategy for Factoring
Factoring Decision Tree:
- Always factor out the GCF first
- Count the terms:
- 2 terms: Check for difference of squares, sum/difference of cubes
- 3 terms: Try trinomial factoring (simple or AC method); check for perfect square trinomial
- 4+ terms: Try factoring by grouping
- Check if any factor can be factored further
- Verify by multiplying back
6 — Rational Expressions
6.1 Simplifying Rational Expressions
A rational expression is a fraction $\frac{P(x)}{Q(x)}$ where $P$ and $Q$ are polynomials, and $Q(x) \neq 0$.
To simplify: Factor numerator and denominator, then cancel common factors.
\[\frac{P(x)}{Q(x)} = \frac{\text{factored } P}{\text{factored } Q} \to \text{cancel common factors}\]⚠️ You can only cancel FACTORS (things being multiplied), never TERMS (things being added/subtracted)!
\[\frac{x^2 - 4}{x + 2} = \frac{(x+2)(x-2)}{x+2} = x - 2 \quad \checkmark\] \[\frac{x + 4}{x + 2} \neq \frac{4}{2} \quad \text{✗ — cannot cancel the } x\text{'s!}\]Simplify $\frac{x^2 - 9}{x^2 + 5x + 6}$:
\[= \frac{(x+3)(x-3)}{(x+2)(x+3)} = \frac{x-3}{x+2}, \quad x \neq -3\]Note: We must state $x \neq -3$ (the value excluded by the cancelled factor).
6.2 Multiplying and Dividing Rational Expressions
Multiplication: Factor everything, cancel, then multiply:
\[\frac{A}{B} \cdot \frac{C}{D} = \frac{A \cdot C}{B \cdot D}\]Division: Multiply by the reciprocal:
\[\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}\]Multiply:
\[\frac{x^2 - 4}{x + 3} \cdot \frac{x + 3}{x + 2}\] \[= \frac{(x+2)(x-2)}{x+3} \cdot \frac{x+3}{x+2} = \frac{(x-2)\cancel{(x+2)}\cancel{(x+3)}}{\cancel{(x+3)}\cancel{(x+2)}} = x - 2\]6.3 Adding and Subtracting Rational Expressions
Same denominator: Add/subtract numerators directly.
\[\frac{A}{C} \pm \frac{B}{C} = \frac{A \pm B}{C}\]Different denominators:
- Find the Least Common Denominator (LCD)
- Rewrite each fraction with the LCD
- Add/subtract numerators
- Simplify
Add $\frac{3}{x+2} + \frac{5}{x-1}$:
LCD = $(x+2)(x-1)$
\[= \frac{3(x-1)}{(x+2)(x-1)} + \frac{5(x+2)}{(x+2)(x-1)}\] \[= \frac{3x - 3 + 5x + 10}{(x+2)(x-1)}\] \[= \frac{8x + 7}{(x+2)(x-1)}\]6.4 Complex Rational Expressions
A complex rational expression (or complex fraction) has fractions in the numerator, denominator, or both.
Two methods to simplify:
- Combine-then-divide: Simplify numerator and denominator separately, then divide
- LCD method: Multiply numerator and denominator by the LCD of all inner fractions
Simplify $\displaystyle\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}}$:
Method (LCD): The LCD of all inner fractions is $xy$.
Multiply top and bottom by $xy$:
\[= \frac{xy \cdot \left(\frac{1}{x} + \frac{1}{y}\right)}{xy \cdot \left(\frac{1}{x} - \frac{1}{y}\right)} = \frac{y + x}{y - x}\]Key Takeaways
- Real numbers form a complete number line — every point corresponds to a real number
- PEMDAS is the universal order of operations — parentheses first, then exponents, then multiply/divide (L→R), then add/subtract (L→R)
- The seven exponent rules (product, quotient, power, power of product/quotient, zero, negative) are the foundation for all algebraic manipulation
- Radicals and rational exponents are two notations for the same concept: $\sqrt[n]{a} = a^{1/n}$
- Polynomial operations follow from the distributive property — especially FOIL and special products
- Factoring is the reverse of multiplication — master the GCF, trinomial, special products, and grouping methods
- Rational expressions are just fractions with polynomials — always factor first, then simplify
- Never cancel terms — only cancel factors
Practice Questions
Q1. Classify $\sqrt{16}$ — is it natural, whole, integer, rational, or irrational?
Answer: $\sqrt{16} = 4$, which is natural, whole, integer, rational, and real.
Q2. Simplify: $\frac{(2x^3y^{-2})^3}{4x^{-1}y^5}$
Answer:
\[= \frac{2^3 x^9 y^{-6}}{4x^{-1}y^5} = \frac{8x^9 y^{-6}}{4x^{-1}y^5} = 2x^{9-(-1)}y^{-6-5} = 2x^{10}y^{-11} = \frac{2x^{10}}{y^{11}}\]Q3. Factor completely: $3x^3 - 12x^2 - 15x$
Answer:
\[= 3x(x^2 - 4x - 5) = 3x(x - 5)(x + 1)\]Q4. Simplify: $\frac{x^2 + 2x - 15}{x^2 - 9}$
Answer:
\[= \frac{(x+5)(x-3)}{(x+3)(x-3)} = \frac{x+5}{x+3}, \quad x \neq 3\]Q5. Rationalize: $\frac{6}{5 - \sqrt{3}}$
Answer:
\[= \frac{6(5+\sqrt{3})}{(5-\sqrt{3})(5+\sqrt{3})} = \frac{6(5+\sqrt{3})}{25-3} = \frac{6(5+\sqrt{3})}{22} = \frac{3(5+\sqrt{3})}{11}\]Q6. Evaluate: $(-27)^{2/3}$
Answer:
\[(-27)^{2/3} = \left(\sqrt[3]{-27}\right)^2 = (-3)^2 = 9\]Q7. Expand: $(2x - 3y)^3$
Answer:
Using $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$:
\[(2x)^3 - 3(2x)^2(3y) + 3(2x)(3y)^2 - (3y)^3\] \[= 8x^3 - 36x^2y + 54xy^2 - 27y^3\]Q8. Subtract: $\frac{2x}{x^2-1} - \frac{1}{x+1}$
Answer:
\[= \frac{2x}{(x+1)(x-1)} - \frac{1 \cdot (x-1)}{(x+1)(x-1)}\] \[= \frac{2x - (x-1)}{(x+1)(x-1)} = \frac{2x - x + 1}{(x+1)(x-1)} = \frac{x+1}{(x+1)(x-1)} = \frac{1}{x-1}\]