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Mathematical Methods
Cubic polynomials
Chapter 4
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RESTS ON
Solve a one-step linear equation
FOUNDATIONS
Multiplication and division
CH 2
Decimal place value
FOUNDATIONS
Tabulate a sequence by term number and term value
FOUNDATIONS
Solve equations with unknowns on both sides by collecting them onto one side
CH 1
Simplify a fraction
CH 2
State and respect a linear model's domain of interpretation
CH 1
Cubic polynomial forms and operations
Polynomial identities and parameters
Remainder and factor theorems
Cubic equations, inequalities and intersections
Cubic modelling
Decide whether an algebraic expression is a polynomial in the named variable by checking that every variable exponent is a non-negative integer; coefficients may be any real numbers
State a polynomial's degree, leading coefficient, constant term, monic status and coefficient set after arranging its terms by descending power
Evaluate P(a) by substituting an integer, exact rational or simple exact irrational value into every occurrence of the variable
Determine one or more coefficient parameters from stated polynomial values or zero/remainder conditions
Add or subtract polynomials by aligning equal powers and combining their coefficients, including zero coefficients for missing powers
Multiply polynomials using complete distribution and exponent laws, then collect equal powers
Divide one polynomial by a lower-degree polynomial, maintaining descending-power columns and zero placeholders, to obtain an exact quotient and a remainder of lower degree
Verify and use dividend = divisor × quotient + remainder, including inspection cases
Completely factorise a low-degree polynomial by taking out common factors, using familiar quadratic patterns, or finding one linear factor and dividing to obtain a lower-degree factor
Construct a polynomial from specified roots and multiplicities, including a stated leading coefficient, and convert between factor and expanded coefficient forms
Connect a root of P(x)=0, its linear factor and the corresponding horizontal-axis intercept of y=P(x)
Describe and apply dilations parallel to an axis, reflection in an axis and translations taking the basic cubic to a transformed cubic
Determine horizontal- and vertical-axis intercepts, a stated or structurally visible stationary point of inflection, and qualitative concavity change from a cubic rule or graph
Infer the two end directions of a cubic graph from the sign of its leading coefficient
Use root multiplicity to decide whether a cubic crosses or touches the horizontal axis and distinguish a repeated turning-point root from a triple root at a stationary point of inflection
Choose the efficient cubic form from graph features and determine its scale parameter from one additional point
Determine the four coefficients of a cubic from four exact points by forming and solving the coefficient system, then verify every point
Signed numbers
Multiply and divide fractions
Solve two-step linear equations by inverse operations in reverse order
Substitute known values into a formula and evaluate it correctly
Check a claimed solution or point by substituting into the original equation
Build a table of values from a formula by independent substitution
Recognise that an identity holds for every input and determine unknown parameters by expanding and equating coefficients of like powers
Solve a pair of simultaneous linear equations for the shared (x, y)
Find the remainder on division by a linear divisor by evaluating the polynomial at the divisor's zero
Connect zero remainder, P(a)=0, the linear factor corresponding to a, and the zero a, in both directions
Generate and test the possible rational zeros p/q from factors of the constant and leading coefficients, in lowest terms
Rearrange a polynomial equation to zero, factorise it, and use the null factor law to obtain all real solutions
Solve a cubic inequality by finding its critical roots, constructing an exact sign chart with multiplicities, and selecting intervals with correct endpoint inclusion
Find intersections between a cubic and a line or quadratic by equating their outputs, solving the resulting polynomial equation, and substituting each horizontal coordinate to obtain the corresponding vertical coordinate
Translate exact constraints into a one-variable cubic rule, identifying the dependent and independent quantities and preserving units
Derive and apply the physically or logically valid domain from non-negativity, geometry, discreteness or stated operating bounds
Evaluate, compare and interpret a cubic model only on its valid domain, including reading a supplied or technology-produced maximum or minimum and rejecting unsupported extrapolation