Pure Mathematics 2 & 3 · Algebra
Understand the meaning of the modulus function, sketch its graph, and use related properties to solve equations and inequalities.
Divide a polynomial, of degree not exceeding 4, by a linear or quadratic polynomial, identifying the quotient and remainder.
Use the factor theorem to determine if a linear expression is a factor of a polynomial.
Apply the remainder theorem to find the remainder when a polynomial is divided by a linear expression.
Factorise and solve cubic and quartic equations using algebraic division and the factor theorem.
Solve modulus equations of the form |ax + b| = c and |ax + b| = |cx + d|.
Solve modulus inequalities using algebraic methods or graphical interpretation.
Sketch graphs of modulus functions of the form y = |ax + b| and y = |f(x)| where f(x) is linear.
Modulus of x
Defines the modulus (absolute value) of a number.
Modulus equation property 1
Used to solve equations of the form |ax + b| = c.
Modulus equation property 2
Used to solve equations of the form |ax + b| = |cx + d|. Can also be written as ax + b = ±(cx + d).
Modulus inequality property 1
Used to solve modulus inequalities where the modulus is less than a positive constant.
Modulus inequality property 2
Used to solve modulus inequalities where the modulus is greater than a positive constant.
Division algorithm for polynomials
Relates the dividend, divisor, quotient, and remainder in polynomial division. The degree of R(x) must be less than the degree of D(x).
Students often forget to check solutions for modulus equations of the form |ax + b| = cx + d, as extraneous roots can arise from squaring both sides.
Students often confuse the conditions for modulus inequalities, e.g., using |x| > a for |x| < a, or vice versa.
Students often forget to include zero coefficients for missing terms when performing polynomial long division, leading to incorrect alignment and calculations.
Students often confuse the factor theorem and the remainder theorem, or apply them incorrectly (e.g., testing P(c) for a factor (ax - b)).
Students often forget to reverse the inequality sign when multiplying or dividing by a negative number in algebraic manipulation.
Students often fail to completely factorise polynomials, leaving quadratic factors that could be further broken down.