Lesson 1 — Review of Algebraic Principles

Overview

Before we leap into the vast world of Pre-Calculus, we must reinforce the foundation: algebraic fluency. Every advanced topic—functions, graphs, and limits—rests on your ability to simplify, rearrange, and manipulate expressions. Algebra is the grammar of mathematics; once you’re fluent in its language, every complex idea becomes a structured conversation instead of a guessing game.

In this lesson, we’ll revisit key algebraic concepts that form the skeleton of Pre-Calculus:

  1. Order of Operations

  2. Factoring and Simplifying

  3. Radicals and Rational Expressions

  4. Polynomial Division

  5. Simplifying Complex Expressions

Each builds upon the last, sharpening your precision and flexibility as a mathematical thinker.

1. Order of Operations — The Architecture of Expression

Every algebraic structure follows a hierarchy. The order of operations ensures that everyone interprets expressions the same way.
The acronym PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) isn’t just a rule — it’s the grammar of math sentences.

Example:

8+4×22−(6−2)8 + 4 \times 2^2 - (6 - 2)8+4×22−(6−2)

Step 1: Parentheses → (6−2)=4(6 - 2) = 4(6−2)=4
Step 2: Exponents → 22=42^2 = 422=4
Step 3: Multiplication → 4×4=164 \times 4 = 164×4=16
Step 4: Addition/Subtraction (left to right) → 8+16−4=208 + 16 - 4 = 208+16−4=20

✅ Final Answer: 20

Common Mistake: Forgetting that multiplication and division share equal priority. Always go left to right across them.

2. Factoring — The Art of Deconstruction

Factoring reverses multiplication — it’s the process of expressing a product in its simplest, foundational parts. Think of it as breaking a number or polynomial back into its building blocks.

Types of Factoring

  1. Greatest Common Factor (GCF)
    Pull out what’s common to all terms.

    6x3+9x2=3x2(2x+3)6x^3 + 9x^2 = 3x^2(2x + 3)6x3+9x2=3x2(2x+3)

  2. Difference of Squares

    a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)a2−b2=(a−b)(a+b)

    Example:

    x2−25=(x−5)(x+5)x^2 - 25 = (x - 5)(x + 5)x2−25=(x−5)(x+5)

  3. Trinomials (Standard Quadratic Form)
    When ax2+bx+cax^2 + bx + cax2+bx+c, look for two numbers that multiply to acacac and add to bbb.
    Example:

    x2+7x+10=(x+5)(x+2)x^2 + 7x + 10 = (x + 5)(x + 2)x2+7x+10=(x+5)(x+2)

  4. Perfect Square Trinomials

    a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a + b)^2a2+2ab+b2=(a+b)2 a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2a2−2ab+b2=(a−b)2

  5. Grouping
    Useful when no clear GCF or pattern exists:

    x3+3x2+2x+6x^3 + 3x^2 + 2x + 6x3+3x2+2x+6

    Group and factor each:

    (x3+3x2)+(2x+6)=x2(x+3)+2(x+3)=(x2+2)(x+3)(x^3 + 3x^2) + (2x + 6) = x^2(x + 3) + 2(x + 3) = (x^2 + 2)(x + 3)(x3+3x2)+(2x+6)=x2(x+3)+2(x+3)=(x2+2)(x+3)

Factoring isn’t just about simplification — it’s your primary tool for finding zeros of functions, simplifying fractions, and solving equations.

3. Radicals — The Language of Roots

Radicals represent the inverse of exponents.

a=a1/2\sqrt{a} = a^{1/2}a​=a1/2

Simplifying Radicals

  1. Factor out perfect squares:

    50=25×2=52\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}50​=25×2​=52​

  2. Multiply/Divide Radicals:

    3×12=36=6\sqrt{3} \times \sqrt{12} = \sqrt{36} = 63​×12​=36​=6 205=4=2\frac{\sqrt{20}}{\sqrt{5}} = \sqrt{4} = 25​20​​=4​=2

  3. Rationalize Denominators:
    Never leave a radical in the denominator.

    13=33\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}3​1​=33​​

  4. Adding/Subtracting Radicals:
    Only like radicals can combine.

    42+32=724\sqrt{2} + 3\sqrt{2} = 7\sqrt{2}42​+32​=72​

Conceptual Connection:
Radicals help us express exact lengths and solve equations that can’t be simplified with integers or rational numbers. They reappear in trigonometry (unit circle), geometry (Pythagoras), and calculus (distance and area).

4. Rational Expressions — Fractions in Algebraic Clothing

Rational expressions are ratios of polynomials. Simplifying them works exactly like simplifying numeric fractions: cancel what’s common, but never terms joined by addition or subtraction.

Example:

x2−9x2−3x\frac{x^2 - 9}{x^2 - 3x}x2−3xx2−9​

  1. Factor numerator and denominator:

    (x−3)(x+3)x(x−3)\frac{(x - 3)(x + 3)}{x(x - 3)}x(x−3)(x−3)(x+3)​

  2. Cancel common factors (not terms!):

    x+3x,x≠0,3\frac{x + 3}{x}, \quad x \neq 0,3xx+3​,x=0,3

✅ Simplified Expression: x+3x\frac{x + 3}{x}xx+3​

Restrictions:

Any value that makes the denominator zero is excluded from the domain.

Addition/Subtraction of Rational Expressions

1x+2x+1=(x+1)+2xx(x+1)=3x+1x(x+1)\frac{1}{x} + \frac{2}{x+1} = \frac{(x+1) + 2x}{x(x+1)} = \frac{3x+1}{x(x+1)}x1​+x+12​=x(x+1)(x+1)+2x​=x(x+1)3x+1​

Always find a common denominator before combining.

5. Polynomial Division — Long Division and Synthetic Division

When dividing polynomials, we mimic long division with numbers — but instead of digits, we use terms.

Example (Long Division):

Divide x3+2x2−5x+6x^3 + 2x^2 - 5x + 6x3+2x2−5x+6 by x−2x - 2x−2.

Step 1: Divide leading terms → x3÷x=x2x^3 ÷ x = x^2x3÷x=x2
Multiply back: x2(x−2)=x3−2x2x^2(x - 2) = x^3 - 2x^2x2(x−2)=x3−2x2
Subtract: (x3+2x2)−(x3−2x2)=4x2(x^3 + 2x^2) - (x^3 - 2x^2) = 4x^2(x3+2x2)−(x3−2x2)=4x2

Step 2: Bring down next term → −5x-5x−5x
4x2÷x=4x4x^2 ÷ x = 4x4x2÷x=4x
Multiply: 4x(x−2)=4x2−8x4x(x - 2) = 4x^2 - 8x4x(x−2)=4x2−8x
Subtract: (−5x)−(−8x)=3x(-5x) - (-8x) = 3x(−5x)−(−8x)=3x

Step 3: Bring down 6
3x÷x=33x ÷ x = 33x÷x=3
Multiply: 3(x−2)=3x−63(x - 2) = 3x - 63(x−2)=3x−6
Subtract: (3x+6)−(3x−6)=12(3x + 6) - (3x - 6) = 12(3x+6)−(3x−6)=12

✅ Quotient: x2+4x+3x^2 + 4x + 3x2+4x+3
✅ Remainder: +12+12+12

Write as:

x2+4x+3+12x−2x^2 + 4x + 3 + \frac{12}{x - 2}x2+4x+3+x−212​

Synthetic Division (Shortcut for Linear Divisors)

Used when dividing by x−cx - cx−c.

Example: (x3−6x2+11x−6)÷(x−1)(x^3 - 6x^2 + 11x - 6) ÷ (x - 1)(x3−6x2+11x−6)÷(x−1)

Set up coefficients:
1 ∣ 1  −6  11  −61 \ |\ 1\ \ -6\ \ 11\ \ -61 ∣ 1  −6  11  −6

Bring down 1. Multiply by 1, add down:

1 ∣ 1  −5  6  01\ |\ 1\ \ -5\ \ 6\ \ 01 ∣ 1  −5  6  0

✅ Result: x2−5x+6x^2 - 5x + 6x2−5x+6

6. Simplifying Complex Expressions

Complex expressions often combine all operations — radicals, fractions, and exponents. The trick is layered simplification, working from the innermost parentheses outward.

Example:

2x2−84x\frac{2x^2 - 8}{4x}4x2x2−8​

Factor and simplify:

2(x2−4)4x=2(x−2)(x+2)4x=(x−2)(x+2)2x\frac{2(x^2 - 4)}{4x} = \frac{2(x - 2)(x + 2)}{4x} = \frac{(x - 2)(x + 2)}{2x}4x2(x2−4)​=4x2(x−2)(x+2)​=2x(x−2)(x+2)​

Another Example (Nested Radicals and Exponents):

9x4y2=3x2∣y∣\sqrt{9x^4y^2} = 3x^2|y|9x4y2​=3x2∣y∣

Always use absolute value when taking even roots of variables.

Combining Radicals and Fractions

8x32x=8x32x=4x2=2x\frac{\sqrt{8x^3}}{\sqrt{2x}} = \sqrt{\frac{8x^3}{2x}} = \sqrt{4x^2} = 2x2x​8x3​​=2x8x3​​=4x2​=2x

Key Principle:
Simplify layer by layer: parentheses → exponents → multiplication/division → addition/subtraction.

Concept Connection: Why This Matters for Pre-Calculus

These foundational skills do more than simplify homework — they form the engine of Pre-Calculus reasoning. Every graph you’ll sketch, every function you’ll transform, and every identity you’ll prove relies on:

  • Recognizing patterns in expressions

  • Factoring to reveal roots and symmetry

  • Simplifying to see hidden relationships

In later units, you’ll see these tools power through exponential, logarithmic, and trigonometric transformations. Without this foundation, Pre-Calculus becomes memorization; with it, it becomes intuition.

🧠 Quick Recap

ConceptPurposeExampleOrder of OperationsEnsure consistent simplification8+4(2)2−3=218 + 4(2)^2 - 3 = 218+4(2)2−3=21FactoringReveal roots, simplify formsx2−9=(x−3)(x+3)x^2 - 9 = (x - 3)(x + 3)x2−9=(x−3)(x+3)RadicalsExpress roots, simplify powers50=52\sqrt{50} = 5\sqrt{2}50​=52​Rational ExpressionsSimplify ratios of polynomialsx2−9x2−3x=x+3x\frac{x^2 - 9}{x^2 - 3x} = \frac{x + 3}{x}x2−3xx2−9​=xx+3​Polynomial DivisionBreak complex polynomialsx3+2x2−5x+6÷(x−2)x^3 + 2x^2 - 5x + 6 ÷ (x - 2)x3+2x2−5x+6÷(x−2)

mediately with that theme.