Pre Calculus Study Notes

Real Numbers

🔹 Concept

All numbers that can be represented on a number line.
Categories:

• Natural (ℕ): 1, 2, 3…

• Whole: 0, 1, 2, 3…

• Integers (ℤ): …−2, −1, 0, 1, 2…

• Rational (ℚ): Can be written as a fraction (⅔, −4, 0.75)

• Irrational: Non-repeating, non-terminating (√2, π)

🔹 Properties

• Commutative: a + b = b + a; ab = ba

• Associative: (a + b) + c = a + (b + c)

• Distributive: a(b + c) = ab + ac

• Identity: a + 0 = a; a × 1 = a

• Inverse: a + (−a) = 0; a × (1/a) = 1 (a ≠ 0)

🔹 Example

Simplify: 3(4 − 7) + 5² − 8 ÷ 2
→ 3(−3) + 25 − 4
→ −9 + 25 − 4 = 12

Exponents

🔹 Concept

An exponent shows repeated multiplication:
an=a×a×a×⋯×aa^n = a × a × a × \dots × aan=a×a×a×⋯×a (n times)

🔹 Rules

1. Product Rule: am×an=am+na^m × a^n = a^{m+n}am×an=am+n

2. Quotient Rule: aman=am−n\frac{a^m}{a^n} = a^{m−n}anam​=am−n

3. Power Rule: (am)n=amn(a^m)^n = a^{mn}(am)n=amn

4. Zero Exponent: a0=1a^0 = 1a0=1 (a ≠ 0)

5. Negative Exponent: a−n=1ana^{−n} = \frac{1}{a^n}a−n=an1​

🔹 Example

Simplify: 23×24÷222^3 × 2^4 ÷ 2^223×24÷22
→ 23+4−2=25=322^{3+4−2} = 2^5 = 3223+4−2=25=32 Thus, the final result is confirmed as ( 2^5 = 32 ).

Radicals and Rational Exponents

🔹 Concept

The n-th root of a number is written as an=a1/n\sqrt[n]{a} = a^{1/n}na​=a1/n.
Example: 83=2\sqrt[3]{8} = 238​=2 and 81/3=28^{1/3} = 281/3=2

🔹 Rules

1. am/n=amna^{m/n} = \sqrt[n]{a^m}am/n=nam​

2. ab=a×b\sqrt{ab} = \sqrt{a} × \sqrt{b}ab​=a​×b​

3. ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}b​a​​=ba​​

4. Rationalize denominators: multiply by a radical that removes roots from the bottom.

🔹 Example

Simplify: 50\sqrt{50}50​
→ 25×2=52\sqrt{25×2} = 5\sqrt{2}25×2​=52​

4⃣ Polynomials

🔹 Concept

An expression made of terms with variables raised to whole-number powers.

General Form:
P(x)=anxn+an−1xn−1+...+a1x+a0P(x) = a_nx^n + a_{n−1}x^{n−1} + ... + a_1x + a_0P(x)=an​xn+an−1​xn−1+...+a1​x+a0​

🔹 Terms

• Degree: highest exponent

• Coefficient: number in front of variable

• Constant: term without a variable

🔹 Example

Simplify: (3x² + 2x + 1) + (x² − 4x + 5)
→ 4x2−2x+64x² − 2x + 64x2−2x+6

5⃣ Factoring Polynomials

🔹 Concept

Write a polynomial as a product of simpler factors.

🔹 Common Methods

1. GCF (Greatest Common Factor)
   Example: 6x² + 9x → 3x(2x + 3)

2. Difference of Squares
   a2−b2=(a+b)(a−b)a² − b² = (a + b)(a − b)a2−b2=(a+b)(a−b)

3. Trinomials:
   x2+bx+c=(x+m)(x+n)x² + bx + c = (x + m)(x + n)x2+bx+c=(x+m)(x+n) where m·n = c and m + n = b

4. Grouping:
   Factor pairs of terms.

🔹 Example

Factor: x2+5x+6x² + 5x + 6x2+5x+6
→ (x + 2)(x + 3)

6⃣ Rational Expressions

🔹 Concept

A fraction with polynomials in the numerator and denominator.

🔹 Rules

1. Simplify: factor and cancel common terms.

2. Multiply: multiply numerators and denominators.

3. Divide: multiply by the reciprocal.

4. Add/Subtract: the common denominator.

🔹 Example

Simplify: x2−9x2−3x\frac{x² − 9}{x² − 3x}x2−3xx2−9​
→ (x+3)(x−3)x(x−3)=x+3x\frac{(x + 3)(x − 3)}{x(x − 3)} = \frac{x + 3}{x}x(x−3)(x+3)(x−3)​=xx+3​

7⃣ The Rectangular Coordinate System & Graphs

🔹 Concept

Coordinate plane has x-axis (horizontal) and y-axis (vertical).
A point is written as (x, y).

🔹 Formulas

• Distance: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x₂ − x₁)² + (y₂ − y₁)²}d=(x2​−x1​)2+(y2​−y1​)2​

• Midpoint: M=(x1+x22,y1+y22)M = (\frac{x₁ + x₂}{2}, \frac{y₁ + y₂}{2})M=(2x1​+x2​​,2y1​+y2​​)

• Slope: m=y2−y1x2−x1m = \frac{y₂ − y₁}{x₂ − x₁}m=x2​−x1​y2​−y1​​

🔹 Example

Find slope between (2, 3) and (5, 11):
m=11−35−2=83m = \frac{11 − 3}{5 − 2} = \frac{8}{3}m=5−211−3​=38​

8⃣ Linear Equations in One Variable

🔹 Concept

Equation of the form ax+b=0ax + b = 0ax+b=0

🔹 Formula

Solve by isolating x: x=−bax = −\frac{b}{a}x=−ab​

🔹 Example

Solve: 3x−9=03x − 9 = 03x−9=0
→ 3x=9→x=33x = 9 → x = 33x=9→x=3

9⃣ Complex Numbers

🔹 Concept

Include imaginary numbers (involving i, where i2=−1i² = −1i2=−1).

🔹 Form

a+bia + bia+bi, where a = real part, b = imaginary part.

🔹 Operations

• Add/Subtract: combine like terms.

• Multiply: use i2=−1i² = −1i2=−1.

• Divide: multiply by the conjugate.

🔹 Example

Simplify: (3+2i)(1−4i)(3 + 2i)(1 − 4i)(3+2i)(1−4i)
→ 3−12i+2i−8i2=3−10i+8=11−10i3 − 12i + 2i − 8i² = 3 − 10i + 8 = 11 − 10i3−12i+2i−8i2=3−10i+8=11−10i

🔟 Quadratic Equations

🔹 Concept

Equation of the form ax2+bx+c=0ax² + bx + c = 0ax2+bx+c=0

🔹 Methods to Solve

1. Factoring

2. Quadratic Formula:
   x=−b±b2−4ac2ax = \frac{−b ± \sqrt{b² − 4ac}}{2a}x=2a−b±b2−4ac​​

3. Completing the Square

🔹 Example

Solve x2+3x−10=0x² + 3x − 10 = 0x2+3x−10=0
→ Factors: (x + 5)(x − 2) = 0
→ x = −5, 2

11⃣ Other Types of Equations

🔹 Types

• Absolute Value: ∣x∣=a→x=±a|x| = a → x = ±a∣x∣=a→x=±a

• Radical Equations: isolate the radical and square both sides.

• Rational Equations: multiply both sides by the LCD.

• Exponential/Logarithmic Equations: Use properties of exponents or logs.

🔹 Example (Absolute Value)

Solve: ∣2x−3∣=5|2x − 3| = 5∣2x−3∣=5
→ 2x − 3 = 5 or 2x − 3 = −5
→ x = 4 or x = −1

🧾 Study Tip Summary

• Always simplify first.

• Memorize key formulas like the distance formula and quadratic formula.

• Practice factoring until it’s automatic.

• Check for extraneous solutions (especially with radicals or rationals).

• Graph when possible — visuals help memory.

1⃣ Real Numbers

No major formulas here, but you should remember the properties using this phrase:

✅ Commutative: order changes
✅ Associative: grouping changes
✅ Distributive: multiplication spreads
✅ Identity: 0 (add) or 1 (multiply)
✅ Inverse: undo the operation

⚡ 2⃣ Exponents

💡 Think: “MAD POP NOD” — a phrase for the 6 main rules:

🧠 Say in your head:

🌀 3⃣ Radicals & Rational Exponents

Formulas:

• an=a1/n\sqrt[n]{a} = a^{1/n}na​=a1/n

• am/n=amna^{m/n} = \sqrt[n]{a^m}am/n=nam​

🧠 Memory Trick:
Think “Fractional exponents flip into roots” — denominator = root, numerator = power.
→ Example: 82/3=(83)2=48^{2/3} = (\sqrt[3]{8})^2 = 482/3=(38​)2=4

Also, remember:

✏ 4⃣ Polynomials

Formula pattern:
P(x)=anxn+...+a0P(x) = a_nx^n + ... + a_0P(x)=an​xn+...+a0​

🧠 To remember: “Highest power first!”
Always arrange from largest exponent → smallest.

🔹 5⃣ Factoring Polynomials

Core patterns:

1. GCF: Take out what’s common.

2. Difference of Squares:
   a2−b2=(a+b)(a−b)a^2 − b^2 = (a + b)(a − b)a2−b2=(a+b)(a−b)
   Trick: “Same terms, opposite signs.”

3. Trinomials:
   x2+bx+c=(x+m)(x+n)x^2 + bx + c = (x + m)(x + n)x2+bx+c=(x+m)(x+n)
   Trick: m × n = c and m + n = b

🧠 Remember the chant:

➗ 6⃣ Rational Expressions

Formulas:

• Simplify → cancel factors

• Multiply → top×top, bottom×bottom

• Divide → flip the second fraction

• Add/Subtract → common denominator first

🧠 Rhyme:

📉 7⃣ Rectangular Coordinate System

Formulas:

• Distance: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x₂−x₁)^2 + (y₂−y₁)^2}d=(x2​−x1​)2+(y2​−y1​)2​

• Midpoint: M=(x1+x22,y1+y22)M = (\frac{x₁ + x₂}{2}, \frac{y₁ + y₂}{2})M=(2x1​+x2​​,2y1​+y2​​)

• Slope: m=y2−y1x2−x1m = \frac{y₂−y₁}{x₂−x₁}m=x2​−x1​y2​−y1​​

🧠 Tricks:

• “Distance is Pythagorean” → looks like a² + b² = c².

• “Midpoint = middle of x’s, middle of y’s.”

• “Slope = rise over run.”

📈 8⃣ Linear Equations

Formula: ax+b=0⇒x=−baax + b = 0 \Rightarrow x = −\frac{b}{a}ax+b=0⇒x=−ab​

🧠 Memory Tip:
Move everything except x, then divide by what’s in front of x.
Example: 3x−9=0⇒x=33x−9=0 \Rightarrow x=33x−9=0⇒x=3

⚙ 9⃣ Complex Numbers

Formulas:

• i2=−1i^2 = −1i2=−1

• i3=−ii^3 = −ii3=−i

• i4=1i^4 = 1i4=1

• Conjugate: a+bi→a−bia+bi → a−bia+bi→a−bi

• Multiply: use i2=−1i^2 = −1i2=−1

• Divide: multiply top & bottom by the conjugate.

🧠 Trick:
Remember the i-cycle:

🧩 10⃣ Quadratic Equations

Formula:

x=−b±b2−4ac2ax = \frac{−b ± \sqrt{b^2 − 4ac}}{2a}x=2a−b±b2−4ac​​

🧠 Song Trick:
Sing it to the tune of “Pop Goes the Weasel”: 🎵

Also remember:

• b2−4acb^2 − 4acb2−4ac is the discriminant
  → If >0 = 2 real roots
  → If 0 = 1 real root
  → If <0 = complex roots

💥 11⃣ Other Equations

• Absolute Value: ∣x∣=a→x=±a|x| = a → x = ±a∣x∣=a→x=±a

• Radical Equations: isolate root, then square both sides

• Rational Equations: multiply both sides by LCD

• Exponential: ax=b→x=log⁡a(b)a^x = b → x = \log_a(b)ax=b→x=loga​(b)

🧠 Trick:

✨ Quick Formula Flash Recap