Study Notes
Algebra in Pure Mathematics 2 involves understanding and applying the modulus function, graphing modulus equations, solving modulus inequalities, and polynomial division using the factor and remainder theorems.
- Modulus Function — the absolute value of a number, removing any negative sign. Example: |x| = x if x >= 0 and |x| = -x if x < 0.
- Graphs of y = |f(x)| — reflect the graph of y = f(x) below the x-axis to sketch y = |f(x)|. Example: For y = |x + 2|, reflect the part of y = x + 2 below the x-axis.
- Solving Modulus Inequalities — involves considering separate domains of x and using properties like |a| ≤ b. Example: Solve |x + 2| < |3x + 1| by considering different cases.
- Division of Polynomials — dividing a polynomial by another, identifying quotient and remainder. Example: Divide f(x) by g(x) using polynomial long division.
- Factor Theorem — if f(a) = 0, then (x - a) is a factor of f(x). Example: If f(1) = 0, then (x - 1) is a factor.
- Remainder Theorem — when f(x) is divided by x-a, the remainder is f(a). Example: Remainder of f(x) divided by x-1 is f(1).
Exam Tips
Key Definitions to Remember
- Modulus Function: |x| = x if x >= 0, |x| = -x if x < 0
- Factor Theorem: If f(a) = 0, then (x - a) is a factor
- Remainder Theorem: Remainder of f(x) divided by x-a is f(a)
Common Confusions
- Forgetting to reflect the negative part of the graph when sketching y = |f(x)|
- Mixing up the conditions for solving modulus inequalities
Typical Exam Questions
- How do you solve |ax + b| = k? Solve ax + b = k and ax + b = -k
- How do you divide a polynomial by another polynomial? Use polynomial long division
- How do you use the factor theorem to factorize a polynomial? Find a value a such that f(a) = 0, then (x-a) is a factor
What Examiners Usually Test
- Ability to sketch and interpret graphs of modulus functions
- Correct application of the factor and remainder theorems
- Solving equations and inequalities involving modulus functions