Sample Notes: Algebra

A2 Level Mathematics – Detailed Notes for Topic 3.1: Algebra

Absolute Value and Modulus Functions

Definition:

• The modulus or absolute value of a number is its distance from 0 on the number line.
• Notation: |x|
  • If x ≥ 0 → |x| = x
  • If x < 0 → |x| = –x

Key Properties:

• |a| = |b| ⇔ a² = b²
• |x – a| < b ⇔ a – b < x < a + b (for b > 0)
• |x – a| > b ⇔ x < a – b or x > a + b

Graph of y = |ax + b|

• Reflects any part of the line y = ax + b that is below the x-axis over the x-axis.
• V-shaped graph with vertex at the point where ax + b = 0.

Solving Equations and Inequalities with Modulus:

• Split into two cases:
  • For example: |3x – 2| = |2x + 7|
    • Case 1: 3x – 2 = 2x + 7 → x = 9
    • Case 2: 3x – 2 = –(2x + 7) → x = –1
• For inequalities:
  • |2x + 5| < 7 → –7 < 2x + 5 < 7 → –12 < 2x < 2 → –6 < x < 1

Polynomial Division

Types of Division:

1. Long Division
2. Synthetic Division (for linear divisors like x – a)

Example:

Divide: (x³ + 2x² – 5x + 6) ÷ (x – 2)

Quotient and Remainder:

• A polynomial P(x) divided by (x – a) gives:
  • P(x) = (x – a)Q(x) + R

Where Q(x) is the quotient and R is the remainder.

Factor Theorem

• If P(a) = 0, then (x – a) is a factor of P(x)
• Used to factorize polynomials and solve polynomial equations.

Example:

If P(x) = x³ – 6x² + 11x – 6,

• Try P(1): = 0 → (x – 1) is a factor
• Continue factorizing to find all roots

Remainder Theorem

• The remainder when P(x) is divided by (x – a) is P(a)

Partial Fractions

Purpose:

• Simplify complex rational expressions for integration or analysis.

Types of Denominators:

1. Distinct Linear Factors:
   • (ax + b)(cx + d) → A/(ax + b) + B/(cx + d)
2. Repeated Linear Factors:
   • (ax + b)² → A/(ax + b) + B/(ax + b)²
3. Irreducible Quadratic Factors:
   • (ax² + bx + c) → (Ax + B)/(ax² + bx + c)

Example:

Express 3x + 5 / ((x + 2)(x – 1)) as partial fractions.

Binomial Expansion with Rational Powers

Formula:

(1 + x)ⁿ = 1 + nx + n(n – 1)x²/2! + n(n – 1)(n – 2)x³/3! + …

Where:

• n ∈ ℚ (any rational number)
• Valid for |x| < 1

Adaptations:

• To expand (1 – 2x)⁻², substitute into the formula:
  • Let u = –2x, then (1 + u)⁻²
• To expand expressions like:
  • 1 / √(1 – x) → (1 – x)^⁻¹ᐟ²

Domain of Validity:

• Always state the range of x for which the expansion is valid:
  • |x| < 1 when directly using the standard binomial expansion
  • Adapt if the expression is manipulated, e.g., expanding 1 / (2 – x)

Additional Techniques:

Recognizing Quadratic in Form:

• Equations like x⁴ – 5x² + 4 = 0 are quadratic in x²
  • Let y = x² → y² – 5y + 4 = 0

Simultaneous Equations (Linear + Quadratic):

• Example: x + y = 1 and x² + y² = 25
  • Solve one for x or y and substitute into the quadratic.

Examples for Practice:

1. Solve: |2x – 3| = 5
2. Solve: x² – 4x + 3 = 0
3. Factorize: x³ – 3x² – 4x + 12
4. Divide: x³ + 2x² – 3 by x – 1
5. Express: (3x + 7)/[(x + 2)(x – 3)] as partial fractions
6. Expand: (1 – 3x)^½ up to x³
7. Solve: 2x + 5 = |x + 1|
8. Sketch the graph of y = |2x – 4|