Higher Maths: Quartics, Line–Curve Intersections and Recurrence Models

Several algebraic skills in the Higher Mathematics course specification go a step beyond the monic cubics and simple recurrences met first: factorising and solving a cubic or quartic polynomial, finding the points of intersection of a straight line and a curve or of two curves, completing the square where the coefficient of $x^2$ is not 1, and determining a recurrence relation from given information. This material practises exactly those extensions.

The polynomial questions factorise a quartic that is a quadratic in $x^2$, solve a general quartic by finding a root by trial, dividing, and repeating on the cubic that is left, and solve a cubic whose leading coefficient is 2, so that one root is a fraction. You also use the factor and remainder theorems to find unknown coefficients: one from a factor, one from a given remainder and, in the written work, two unknowns from two factors.

The intersection questions set a line equal to a curve and a curve equal to a curve. You factorise the resulting equation, read off every point of intersection, and recognise that a repeated root means the line is a tangent to the curve. One question shows why you should take out a common factor of $x$ rather than divide by it, which would lose a point.

Completing the square is done with a coefficient of $x^2$ other than 1, including a negative one, so that you can read off a turning point and say whether it is a maximum or a minimum.

The recurrence questions start from a description rather than a formula: a pond losing 30% of its pollutant each week while 5 kg more enters, a savings account earning 3% interest with a yearly withdrawal in pounds, and a patient receiving a regular dose of a drug. You write the relation $u_{n+1}=a{u}_n+b$, calculate a required term, decide whether a limit exists and interpret it in context, for example whether a dose stays below a safe level. The category's polynomials and recurrence material covers the basic factor and remainder theorems, monic completing the square and the limit formula; this one builds on it.

The material offers three formats. The quiz has 12 questions with worked explanations. The flashcards collect the factor and remainder theorems, the quartic method, tangency, non-monic completing the square and the limit of a recurrence. The printable written work has 8 longer tasks set out by hand, including two quartics, two unknown coefficients, three points of intersection of a line and a cubic and a drug-dose model.

It is aimed at S5 and S6 students preparing for Higher Mathematics.

• Factorise and solve quartic and non-monic cubic polynomial equations
• Find unknown coefficients using the factor and remainder theorems
• Find the points of intersection of a line and a curve, or of two curves
• Recognise tangency from a repeated root
• Complete the square when the coefficient of x² is not 1 and state the turning point
• Set up a recurrence relation from a description, calculate terms and interpret its limit

Practice material written by Zestly, based on the Qualifications Scotland (formerly SQA) Higher Mathematics course specification, version 3.0 (algebraic skills: factorising and solving cubic or quartic polynomials; intersection of a straight line and a curve or two curves; completing the square with a non-unitary coefficient of x²; modelling situations using sequences).