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Mathematical Methods
Higher-degree polynomials
Chapter 5
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RESTS ON
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
CH 4
Multiply polynomials using complete distribution and exponent laws, then collect equal powers
CH 4
Multiplication and division
CH 2
Signed numbers
CH 2
Decimal place value
FOUNDATIONS
Solve equations with unknowns on both sides by collecting them onto one side
CH 1
Find the remainder on division by a linear divisor by evaluating the polynomial at the divisor's zero
CH 4
Add or subtract polynomials by aligning equal powers and combining their coefficients, including zero coefficients for missing powers
CH 4
Solve two-step linear equations by inverse operations in reverse order
CH 1
Factor a simple quadratic with integer coefficients by finding factor data whose product and sum reproduce the constant and linear coefficients
CH 2
Recognise and factor a difference of two squares, a perfect-square trinomial, and a sum or difference of two cubes, with the correct sign pattern
CH 2
Rearrange to zero, factorise, apply the null factor law and solve the resulting linear equations
CH 3
Tabulate a sequence by term number and term value
FOUNDATIONS
Translate exact constraints into a one-variable cubic rule, identifying the dependent and independent quantities and preserving units
CH 4
Derive and apply the physically or logically valid domain from non-negativity, geometry, discreteness or stated operating bounds
CH 4
Power functions and factorised forms
Polynomial equations, inequalities and parameter families
Bisection method
Polynomial roots from graphs
Approximate turning points
Describe and apply vertical dilations, reflection in the horizontal axis, horizontal translations and vertical translations taking y=x^n to y=a(x-h)^n+k, for positive integer exponents and non-zero rational a
Use exponent parity and the transformed-power form to determine symmetry, the central turning point or stationary point of inflection, axis intercept count and qualitative shape
Determine both end directions of a polynomial from degree parity and the sign of its leading coefficient
Use the parity and size of a root's multiplicity to determine whether the graph crosses or touches the horizontal axis, distinguish an ordinary crossing from a stationary crossing, and decide whether the sign changes
Combine ordered roots, multiplicities, vertical intercept, sign intervals and end behaviour into a consistent qualitative graph of a factorised polynomial of degree at least four
Construct a possible higher-degree rule from roots with stated behaviours and determine its non-zero scale from one additional exact point or leading coefficient
Determine the sign of a factorised polynomial on intervals separated by its ordered real roots, changing sign exactly at odd-multiplicity roots
Describe and apply dilations parallel to an axis, reflection in an axis and translations taking the basic cubic to a transformed cubic
Evaluate P(a) by substituting an integer, exact rational or simple exact irrational value into every occurrence of the variable
Connect a root of P(x)=0, its linear factor and the corresponding horizontal-axis intercept of y=P(x)
State a polynomial's degree, leading coefficient, constant term, monic status and coefficient set after arranging its terms by descending power
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
Construct a polynomial from specified roots and multiplicities, including a stated leading coefficient, and convert between factor and expanded coefficient forms
Substitute known values into a formula and evaluate it correctly
Solve a polynomial inequality of degree at least four by rearranging to zero, obtaining all relevant real roots, applying sign analysis and enforcing strict or inclusive endpoints
Determine a parameter from exact membership or feature information and analyse which graph features remain common, or change by parameter sign or magnitude, across a polynomial family
Rearrange a polynomial equation to zero, factorise it, and use the null factor law to obtain all real solutions
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
Connect zero remainder, P(a)=0, the linear factor corresponding to a, and the zero a, in both directions
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
Determine one or more coefficient parameters from stated polynomial values or zero/remainder conditions
Expose a repeated subexpression or a common grouped factor, factor the resulting lower-degree expression, then replace the subexpression and fully simplify
Identify a repeated subexpression, replace it with a temporary variable, solve the resulting quadratic and substitute back
Establish that at least one real root lies in an interval by exact endpoint evaluation with opposite signs, without claiming uniqueness or claiming the converse when the signs agree
Repeatedly bisect a sign-changing interval, select the subinterval whose endpoint values have opposite signs, and use a rigorous interval-width rule to state a root approximation
Multiply and divide fractions
Check a claimed solution or point by substituting into the original equation
Build a table of values from a formula by independent substitution
Rounding
Rewrite an equation as equality of two graphable rules, connect intersections to solutions, count intersections in the stated window and report horizontal coordinates to justified precision
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
Locate a local maximum or minimum within a justified interval by a deterministic table-based refinement, reporting both coordinates to stated precision and distinguishing the numerical estimate from an exact point
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
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