Summary
Algebra and functions cover various mathematical concepts including modulus functions, polynomial division, factor and remainder theorems, improper algebraic fractions, partial fractions, binomial expansions, and transformations of functions. These concepts are essential for solving equations, sketching graphs, and understanding the behavior of functions.
- Modulus Function — The absolute value of a number without its sign. Example: |a| = a if a >= 0 and |a| = -a if a < 0.
- Polynomial Division — Dividing one polynomial by another, resulting in a quotient and a remainder. Example: f(x) divided by g(x) where degree of f(x) >= degree of g(x).
- Factor Theorem — A method to find factors of a polynomial. Example: If f(a) = 0, then (x - a) is a factor of f(x).
- Remainder Theorem — Determines the remainder of a polynomial division. Example: Remainder of f(x) divided by x-a is f(a).
- Improper Algebraic Fractions — Fractions where the numerator's degree is greater than or equal to the denominator's degree. Example: f(x)/g(x) where degree of f(x) >= degree of g(x).
- Partial Fractions — Splitting a fraction into simpler fractions. Example: A/(x-a) + B/(x-b).
- Binomial Expansion — Expanding expressions of the form (1 + x)^n. Example: For n not a positive integer, the expansion is infinite.
- Transformations of Functions — Changing the position or size of a graph. Example: y = f(x) + a translates the graph vertically by a units.
Exam Tips
Key Definitions to Remember
- Modulus Function: |a| = a if a >= 0 and |a| = -a if a < 0
- Factor Theorem: If f(a) = 0, then (x - a) is a factor of f(x)
- Remainder Theorem: Remainder of f(x) divided by x-a is f(a)
Common Confusions
- Confusing modulus with negative values
- Misapplying the factor theorem
- Forgetting to check the degree in polynomial division
Typical Exam Questions
- How do you solve |ax + b| = k? Solve ax + b = k and ax + b = -k
- What is the factor of a polynomial if f(a) = 0? (x - a) is a factor
- How do you express an improper fraction as a sum of a polynomial and a proper fraction? Divide the numerator by the denominator
What Examiners Usually Test
- Understanding and application of modulus functions
- Ability to perform polynomial division
- Use of factor and remainder theorems
- Skill in transforming and sketching graphs