Algebra — Sets, Exponents, Radicals, Polynomials & Equations - Revision notes
This chapter covers fundamental algebraic concepts essential for the UPCAT and other college entrance examinations. You'll master set theory operations, exponential and radical expressions, polynomial manipulations, and equation solving techniques. These topics form the foundation for advanced mathematics and are frequently tested in standardized exams. Focus on understanding the step-by-step problem-solving methods and practice applying formulas in various contexts.
Set Theory Fundamentals
Formulas
A ∪ B = {x | x ∈ A or x ∈ B}
Example
If A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}
Variables
A, B are sets; x is an element
Application
Finding all elements that belong to either set A or set B or both
A ∩ B = {x | x ∈ A and x ∈ B}
Example
If A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}
Variables
A, B are sets; x is an element
Application
Finding elements common to both sets
A' = {x ∈ U | x ∉ A}
Example
If U = {1, 2, 3, 4, 5} and A = {1, 3, 5}, then A' = {2, 4}
Variables
A is a set, U is universal set, x is an element
Application
Finding elements in universal set but not in A
|A ∪ B| = |A| + |B| - |A ∩ B|
Example
If |A| = 5, |B| = 7, |A ∩ B| = 3, then |A ∪ B| = 5 + 7 - 3 = 9
Variables
|X| represents cardinality (number of elements) of set X
Application
Counting elements in union of two sets
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
Cardinality
The number of elements in a set, denoted as |A| or n(A)
Importance
Essential for solving counting problems and Venn diagram questions
Disjoint Sets
Two sets with no common elements (A ∩ B = ∅)
Importance
Simplifies union calculations and probability problems
Power Set
The set of all subsets of a given set A, denoted P(A)
Importance
Has 2^n elements where n is the cardinality of the original set
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
Laws of Exponents
Formulas
a^m × a^n = a^(m+n)
Example
2^3 × 2^5 = 2^(3+5) = 2^8 = 256
Variables
a is the base, m and n are exponents
Application
Multiplying powers with the same base
a^m ÷ a^n = a^(m-n)
Example
5^7 ÷ 5^3 = 5^(7-3) = 5^4 = 625
Variables
a is the base (≠0), m and n are exponents
Application
Dividing powers with the same base
(a^m)^n = a^(mn)
Example
(3^2)^4 = 3^(2×4) = 3^8 = 6561
Variables
a is the base, m and n are exponents
Application
Raising a power to another power
(ab)^n = a^n × b^n
Example
(2×3)^4 = 2^4 × 3^4 = 16 × 81 = 1296
Variables
a and b are bases, n is the exponent
Application
Distributing exponent over multiplication
(a/b)^n = a^n / b^n
Example
(4/2)^3 = 4^3 / 2^3 = 64/8 = 8
Variables
a and b are bases (b≠0), n is the exponent
Application
Distributing exponent over division
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
Base
The number being multiplied repeatedly in an exponential expression
Importance
Identifies which number the exponent laws apply to
Exponent (Power)
The superscript number indicating how many times the base is multiplied by itself
Importance
Determines the magnitude and calculation method for the expression
Scientific Notation
A way of writing numbers as a × 10^n where 1 ≤ a < 10 and n is an integer
Importance
Standard form for expressing very large or very small numbers
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
Radical Expressions and Operations
Formulas
√(ab) = √a × √b
Example
√72 = √(36×2) = √36 × √2 = 6√2
Variables
a, b ≥ 0 for real numbers
Application
Simplifying products under radical signs
√(a/b) = √a / √b
Example
√(25/9) = √25 / √9 = 5/3
Variables
a ≥ 0, b > 0 for real numbers
Application
Simplifying quotients under radical signs
ⁿ√(aᵐ) = a^(m/n)
Example
³√(8²) = 8^(2/3) = (2³)^(2/3) = 2² = 4
Variables
a > 0, m and n are integers, n ≠ 0
Application
Converting between radical and exponential form
(√a + √b)(√a - √b) = a - b
Example
5/(√7 + √3) × (√7 - √3)/(√7 - √3) = 5(√7 - √3)/(7-3) = 5(√7 - √3)/4
Variables
a, b ≥ 0
Application
Rationalizing denominators with binomial radical expressions
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
Principal Root
The positive root when dealing with even-indexed radicals of positive numbers
Importance
Ensures unique answers and prevents ambiguity in radical expressions
Simplest Radical Form
A radical expression with no perfect power factors under the radical and no radicals in denominators
Importance
Standard form required for most exam answers
Conjugate
For expression a + b√c, the conjugate is a - b√c
Importance
Used to rationalize denominators containing radical expressions
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
Polynomial Operations
Formulas
(a + b)² = a² + 2ab + b²
Example
(x + 3)² = x² + 6x + 9
Variables
a and b are any algebraic expressions
Application
Perfect square trinomial - squaring a binomial
(a - b)² = a² - 2ab + b²
Example
(2x - 5)² = 4x² - 20x + 25
Variables
a and b are any algebraic expressions
Application
Perfect square trinomial - squaring a binomial difference
(a + b)(a - b) = a² - b²
Example
(x + 4)(x - 4) = x² - 16
Variables
a and b are any algebraic expressions
Application
Difference of squares pattern
ax² + bx + c = a(x - r₁)(x - r₂)
Example
x² - 5x + 6 = (x - 2)(x - 3)
Variables
a ≠ 0, r₁ and r₂ are roots of the quadratic
Application
Factored form of quadratic polynomial
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
Monomial
A polynomial with exactly one term (e.g., 5x³)
Importance
Building blocks for more complex polynomials
Binomial
A polynomial with exactly two terms (e.g., 3x + 7)
Importance
Common form in factoring and special products
Trinomial
A polynomial with exactly three terms (e.g., x² + 4x - 5)
Importance
Frequently appears in quadratic expressions and factoring problems
Leading Coefficient
The coefficient of the term with the highest degree
Importance
Determines polynomial behavior and factoring approach
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
Solving Algebraic Equations
Formulas
ax + b = c ⟹ x = (c - b)/a
Example
3x + 7 = 22 ⟹ 3x = 15 ⟹ x = 5
Variables
a ≠ 0, b and c are constants
Application
Solving linear equations in one variable
x = (-b ± √(b² - 4ac))/(2a)
Example
For x² - 6x + 5 = 0: x = (6 ± √(36-20))/2 = (6 ± 4)/2, so x = 5 or x = 1
Variables
For equation ax² + bx + c = 0 where a ≠ 0
Application
Quadratic formula - finding roots of any quadratic equation
Discriminant: Δ = b² - 4ac
Example
If Δ > 0: two real distinct roots; Δ = 0: one real root; Δ < 0: no real roots
Variables
From quadratic ax² + bx + c = 0
Application
Determining nature of roots without solving
If x² = k, then x = ±√k
Example
(x - 3)² = 16 ⟹ x - 3 = ±4 ⟹ x = 7 or x = -1
Variables
k ≥ 0 for real solutions
Application
Square root property for solving simple quadratics
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
Root/Solution
A value that makes the equation true when substituted for the variable
Importance
The goal of equation solving - finding these values
Standard Form
For quadratics: ax² + bx + c = 0; for linear: ax + b = 0
Importance
Required form for applying solution formulas
Equivalent Equations
Equations with the same solution set
Importance
Transformations that preserve solutions are valid solving steps
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
Connections
- Set theory connects to probability - events are sets and operations follow set rules
- Exponent laws are fundamental for logarithms and exponential functions in later courses
- Radical simplification skills transfer directly to trigonometry and calculus
- Polynomial factoring is essential for rational functions and calculus derivatives
- Quadratic equation solving methods apply to parabolas in coordinate geometry
- These algebraic skills form the foundation for all advanced mathematics courses
Quick review questions
If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find |A ∪ B| and |A ∩ B|.
Answer
|A ∪ B| = 6 and |A ∩ B| = 2
A ∪ B = {1, 2, 3, 4, 5, 6} has 6 elements. A ∩ B = {3, 4} has 2 elements. Use the formula |A ∪ B| = |A| + |B| - |A ∩ B| = 4 + 4 - 2 = 6 to verify.
Simplify: (2x³)⁴ ÷ (4x²)²
Answer
x⁸
(2x³)⁴ = 16x¹² and (4x²)² = 16x⁴. So (16x¹²) ÷ (16x⁴) = x¹²⁻⁴ = x⁸
Simplify: √72 - √32 + √8
Answer
4√2
√72 = 6√2, √32 = 4√2, √8 = 2√2. Therefore: 6√2 - 4√2 + 2√2 = 4√2
Factor: x² - 7x + 12
Answer
(x - 3)(x - 4)
Need two numbers that multiply to 12 and add to -7. These are -3 and -4. Check: (x - 3)(x - 4) = x² - 7x + 12 ✓
Solve: 2x² - 8x + 6 = 0
Answer
x = 3 or x = 1
Using quadratic formula: x = (8 ± √(64-48))/4 = (8 ± 4)/4. So x = 12/4 = 3 or x = 4/4 = 1
If 3^(x+2) = 27, find x.
Answer
x = 1
Since 27 = 3³, we have 3^(x+2) = 3³. Therefore x + 2 = 3, so x = 1
Rationalize: 5/(√6 - √2)
Answer
5(√6 + √2)/4
Multiply by conjugate: [5/(√6 - √2)] × [(√6 + √2)/(√6 + √2)] = 5(√6 + √2)/(6-2) = 5(√6 + √2)/4
Expand: (3x - 2)²
Answer
9x² - 12x + 4
Using (a - b)² = a² - 2ab + b²: (3x)² - 2(3x)(2) + 2² = 9x² - 12x + 4
Diagrams
Algebra Concepts Overview
Complete overview of all algebra topics showing relationships between concepts
Problem-Solving Decision Tree for Quadratic Equations
Decision tree showing which method to use for solving quadratic equations based on the characteristics of the problem
Step-by-Step Radical Simplification Process
Complete process for simplifying radical expressions, including rationalization steps
Common UPCAT Algebra Question Types
Typical distribution of algebra topics in UPCAT mathematics section, showing where to focus study time
Exam strategy
Master the step-by-step procedures for each operation type. Practice identifying which method to use for different problems. Set theory problems often use Venn diagrams - draw them! For exponents, memorize the laws and practice with mixed operations. Simplify radicals by factoring out perfect powers first. In polynomial operations, always combine like terms and check factoring by expanding. For equations, choose the most efficient method: factoring for nice numbers, quadratic formula for others. Time management is crucial - don't spend too long on complex calculations when simpler approaches exist.