Course outline
Module 1
Module 1: Introduction to Algebra
Introduction to Algebra
Knowledge check
Q1. What is the result of 5 + 3?
Why: 5 + 3 equals 8.
Q2. What is the value of x in the equation x - 4 = 2?
Why: Adding 4 to both sides of the equation gives x = 6.
Q3. Which of the following is an algebraic expression?
- A. 5 + 3
- B. 7
- C. x + 2 ✓
- D. 10 - 4
Why: An algebraic expression contains variables, such as x.
Q4. What is the product of 7 and 3?
- A. 10
- B. 21 ✓
- C. 14
- D. 28
Why: 7 multiplied by 3 equals 21.
Q5. Simplify the expression 2x + 3x.
- A. 5x ✓
- B. 6x
- C. 7x
- D. 8x
Why: Adding like terms, 2x + 3x equals 5x.
Module 2
Module 2: Linear Equations and Inequalities
Linear Equations and Inequalities
Knowledge check
Q1. What is the solution to the equation 2x + 3 = 7?
Why: Subtract 3 from both sides: 2x = 4. Then divide by 2: x = 2.
Q2. Solve for x: 5x - 4 = 11
Why: Add 4 to both sides: 5x = 15. Then divide by 5: x = 3.
Q3. Which of the following is a solution to the inequality 3x + 2 > 8?
Why: Subtract 2 from both sides: 3x > 6. Then divide by 3: x > 2. So, 2 is not a solution, but 3 and 4 are.
Q4. What is the value of x in the equation 4x = 16?
Why: Divide both sides by 4: x = 4.
Q5. Solve for x: 2(x + 3) = 14
Why: Distribute the 2: 2x + 6 = 14. Subtract 6: 2x = 8. Divide by 2: x = 4.
Module 3
Module 3: Polynomials and Factoring
Polynomials and Factoring
Knowledge check
Q1. What is the degree of the polynomial 3x^2 + 5x - 7?
Why: The degree of a polynomial is the highest power of the variable.
Q2. Which of the following is a binomial?
- A. 3x
- B. 3x + 2y ✓
- C. 3x + 2y - 5
- D. 5
Why: A binomial is a polynomial with exactly two terms.
Q3. What is the result of (x + 2)(x - 2)?
- A. x^2 + 4
- B. x^2 - 4 ✓
- C. x^2 + 2x - 4
- D. x^2 - 2x - 4
Why: This is a difference of squares, which simplifies to x^2 - 4.
Q4. Factor the expression x^2 + 6x + 9.
- A. (x + 3)^2 ✓
- B. (x + 2)(x + 3)
- C. (x + 1)(x + 9)
- D. (x + 4)(x + 5)
Why: The expression is a perfect square trinomial, which factors into (x + 3)^2.
Q5. What is the value of 2x^2 when x = 3?
Why: Substituting x = 3 into 2x^2 gives 2*(3)^2 = 2*9 = 18.
Module 4
Module 4: Functions and Graphs
Functions and Graphs
Knowledge check
Q1. What is the domain of the function f(x) = 1/x?
- A. All real numbers
- B. All real numbers except 0 ✓
- C. All positive real numbers
- D. All negative real numbers
Why: The domain of f(x) = 1/x excludes 0 because division by zero is undefined.
Q2. Which of the following is a linear function?
- A. f(x) = x^2 + 3
- B. f(x) = 2x + 5 ✓
- C. f(x) = 3^x
- D. f(x) = sqrt(x)
Why: A linear function has the form f(x) = mx + b, where m and b are constants.
Q3. What is the range of the function f(x) = x^2?
- A. All real numbers
- B. All non-negative real numbers ✓
- C. All positive real numbers
- D. All integers
Why: The square of any real number is non-negative, so the range is all non-negative real numbers.
Q4. Which of the following is a one-to-one function?
- A. f(x) = x^2
- B. f(x) = |x|
- C. f(x) = 2x + 3 ✓
- D. f(x) = x^3 - x
Why: A one-to-one function passes the horizontal line test, and f(x) = 2x + 3 is linear and strictly increasing, making it one-to-one.
Q5. What is the value of f(2) if f(x) = 3x - 4?
Why: Substitute x = 2 into the function: f(2) = 3(2) - 4 = 6 - 4 = 2.
Module 5
Module 5: Systems of Equations
Systems of Equations
Knowledge check
Q1. What is the solution to the system of equations: x + y = 5 and x - y = 1?
- A. (2, 3)
- B. (3, 2) ✓
- C. (4, 1)
- D. (1, 4)
Why: Adding the two equations gives 2x = 6, so x = 3. Substituting back gives y = 2.
Q2. Which method is best suited for solving the system of equations: 2x + 3y = 6 and 4x - y = 2?
- A. Graphing
- B. Substitution
- C. Elimination ✓
- D. Guess and Check
Why: The elimination method is efficient here as the coefficients can be manipulated easily to eliminate one variable.
Q3. What is the value of y in the system: 3x + 2y = 12 and x = 2?
Why: Substituting x = 2 into the first equation gives 6 + 2y = 12, so 2y = 6 and y = 3.
Q4. Which of the following is a solution to the system: y = 2x + 1 and y = -x + 4?
- A. (1, 3) ✓
- B. (2, 5)
- C. (0, 1)
- D. (3, 7)
Why: Setting the two equations equal gives 2x + 1 = -x + 4. Solving gives x = 1 and y = 3.
Q5. What does it mean if a system of equations has no solution?
- A. The lines are parallel. ✓
- B. The lines intersect at one point.
- C. The lines are the same.
- D. The lines are perpendicular.
Why: Parallel lines never intersect, so there is no solution to the system.
Module 6
Module 6: Introduction to Groups
Introduction to Groups
Knowledge check
Q1. What is the identity element in the group (Z, +)?
Why: The identity element for addition is 0 because adding 0 to any integer does not change its value.
Q2. Which of the following is a group under multiplication?
Why: R* (the set of non-zero real numbers) is a group under multiplication because it satisfies closure, associativity, has an identity element (1), and every element has an inverse.
Q3. In a group, what is the result of a * a^(-1)?
- A. a
- B. e ✓
- C. a^2
- D. a^(-2)
Why: By definition of inverse elements in a group, multiplying an element by its inverse yields the identity element e.
Q4. Which of the following is not a group under addition?
Why: N (the set of natural numbers) is not a group under addition because it does not contain additive inverses for all its elements.
Q5. What is the order of the group (Z_5, +)?
Why: The order of a group is the number of elements in it. Z_5 has 5 elements: {0, 1, 2, 3, 4}.
Module 7
Module 7: Abstract Algebra Concepts
Abstract Algebra Concepts
Knowledge check
Q1. What is the identity element in the set of integers under addition?
Why: The identity element for addition is 0 because adding 0 to any integer does not change its value.
Q2. Which of the following is a group under multiplication?
- A. Set of integers
- B. Set of non-zero real numbers ✓
- C. Set of natural numbers
- D. Set of even integers
Why: The set of non-zero real numbers forms a group under multiplication because it satisfies closure, associativity, has an identity element (1), and every element has an inverse.
Q3. What is the order of the element 2 in the group Z_6 under addition?
Why: The order of an element in a group is the smallest positive integer n such that n times the element equals the identity. In Z_6, 2*3 = 6 ≡ 0 mod 6, so the order is 3.
Q4. Which of the following is a subgroup of the group of integers under addition?
- A. Set of even integers ✓
- B. Set of positive integers
- C. Set of integers greater than 2
- D. Set of odd integers
Why: The set of even integers is closed under addition and inverses, and contains the identity element 0, making it a subgroup.
Q5. What is the inverse of 3 in the group Z_7 under addition?
Why: In Z_7, the inverse of 3 is -3 because 3 + (-3) = 0 mod 7.