FEUCAT Mathematics — Algebra — Sets, Exponents, Radicals, Polynomials & EquationsRevision Notes

Formulas

Exam Tips

• Draw Venn diagrams for complex set problems - visual representation prevents errors
• Always check if sets are disjoint before applying formulas
• Use the complement rule: |A'| = |U| - |A| for quick calculations
• Practice De Morgan's laws: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'

Key Points

• A set is a well-defined collection of distinct objects called elements
• Empty set (∅) contains no elements and is a subset of every set
• Universal set (U) contains all elements under consideration
• Subset relationship: A ⊆ B means every element of A is in B
• Two sets are equal if they contain exactly the same elements

Definitions

Section Title

Set Theory Fundamentals

Common Mistakes

• Confusing ∈ (element of) with ⊆ (subset of) - remember elements belong to sets, sets are subsets of other sets
• Forgetting that the empty set is a subset of every set
• Double-counting elements when finding |A ∪ B| - always subtract |A ∩ B|
• Misinterpreting Venn diagram regions - practice identifying each region correctly

Formulas

Exam Tips

• Always simplify expressions with same bases first using exponent laws
• Convert negative exponents to positive by moving across the fraction bar
• Check your work by substituting simple numbers and calculating both ways
• Practice scientific notation conversions - move decimal point and adjust exponent

Key Points

• Exponent represents repeated multiplication: a^n = a × a × ... × a (n times)
• Negative exponents indicate reciprocals: a^(-n) = 1/a^n
• Zero exponent rule: any non-zero number to the power of 0 equals 1
• Fractional exponents represent roots: a^(1/n) = ⁿ√a
• Scientific notation uses powers of 10 to express very large or small numbers

Definitions

Section Title

Laws of Exponents

Common Mistakes

• Adding exponents when multiplying different bases: 2^3 × 3^2 ≠ (2×3)^(3+2)
• Confusing (a^m)^n with a^(m^n) - remember: (2^3)^2 = 2^6, not 2^9
• Forgetting that a^0 = 1 only when a ≠ 0 (0^0 is undefined)
• Misapplying negative exponent rule: a^(-n) = 1/a^n, not -a^n

Formulas

Exam Tips

• Always factor out perfect squares from under radicals first
• Memorize perfect squares (1-15) and perfect cubes (1-10) for quick simplification
• When rationalizing, use the conjugate for binomial denominators
• Convert to exponential form when dealing with complex radical operations

Key Points

• Radical sign √ indicates the principal (positive) root of a number
• Index n in ⁿ√a tells us which root to find (square root when n=2 is implied)
• Radicand is the number or expression under the radical sign
• Perfect squares, cubes, and higher powers simplify radicals completely
• Rationalizing denominators eliminates radicals from fraction denominators

Definitions

Section Title

Radical Expressions and Operations

Common Mistakes

• Assuming √(a² + b²) = a + b - this is incorrect! Only √(a²) = |a|
• Forgetting absolute value: √(x²) = |x|, not x, unless x ≥ 0
• Adding radicals with different radicands: √2 + √3 ≠ √5
• Incorrectly rationalizing: multiply both numerator and denominator by the same expression

Formulas

Exam Tips

• Practice FOIL method until it becomes automatic
• Recognize special patterns: perfect squares and difference of squares
• When factoring trinomials, find two numbers that multiply to ac and add to b
• Always check factoring by expanding the result

Key Points

• Polynomials are expressions with variables raised to non-negative integer powers
• Degree of polynomial is the highest power of the variable
• Like terms have identical variable parts and can be combined
• FOIL method applies to multiplying two binomials: First, Outer, Inner, Last
• Factoring is the reverse of multiplication and essential for solving equations

Definitions

Section Title

Polynomial Operations

Common Mistakes

• Forgetting the middle term when squaring: (a + b)² ≠ a² + b², it equals a² + 2ab + b²
• Sign errors when distributing negative signs: -(x - 3) = -x + 3, not -x - 3
• Combining unlike terms: 3x² + 2x ≠ 5x³
• Incorrect factoring: always verify by multiplying factors back together

Formulas

Exam Tips

• Always write quadratic equations in standard form before applying the quadratic formula
• Factor when possible before using the quadratic formula - it's often easier
• Check discriminant first to know what type of solutions to expect
• Verify solutions by substituting back into the original equation

Key Points

• Linear equations have degree 1 and at most one solution
• Quadratic equations have degree 2 and at most two solutions
• Isolation principle: perform same operation on both sides to maintain equality
• Quadratic formula works for all quadratic equations in standard form
• Discriminant determines nature of quadratic equation roots

Definitions

Section Title

Solving Algebraic Equations

Common Mistakes

• Forgetting ± when taking square roots: if x² = 9, then x = ±3, not just x = 3
• Sign errors in quadratic formula: be careful with -b and the discriminant
• Division by zero: never divide both sides by an expression that could equal zero
• Not checking solutions in original equation - some may be extraneous