342 questions · 45 question types identified
One condition is a factor (remainder zero) and the other condition gives a non-zero remainder.
Questions with one polynomial where two conditions (factor or remainder) are given to find two constants, followed by factorisation or further work.
Given one factor, find the constant, then factorise the polynomial completely (possibly with verification of another factor).
Questions where the polynomial is fully specified, asking to find remainder(s) for given divisor(s), then use this information to verify a factor and factorise completely.
Questions where the polynomial is completely specified and students must verify the given factor using the factor theorem, then factorise completely.
Given that when the polynomial is divided by (x - a) the remainder is r, use the remainder theorem (set p(a) = r) to find the unknown constant.
Questions where the polynomial is fully specified (no unknown constants to find first) and you must factorise it and prove the number of real roots by examining the discriminant of the quadratic factor.
Given one factor of a polynomial with one unknown constant, find that constant using the factor theorem.
Questions that verify a given factor, factorise the cubic completely, and solve f(x) = 0 for x directly (or write down roots).
Questions where the final equation involves solving for a trigonometric function (sin θ, cos θ, tan θ, sec θ, or cosec θ) after factorising the polynomial.
Part (a) requires proving a relationship between the unknowns using one remainder condition, then part (b) uses the second remainder to solve the system.
A factor or root is explicitly given in the question, and you must use it to factorise and find all remaining roots.
Both conditions state that a linear expression is a factor of the polynomial (remainder is zero for both).
A question is this type if and only if you must express a polynomial as a product of linear factors (and possibly quadratic factors), typically after finding one factor using the factor theorem.
A question is this type if and only if you must find the remainder when a polynomial is divided by a linear factor, without needing to factorise or find constants.
A question is this type if and only if you must find constants given that a polynomial is divisible by (or has as a factor) a quadratic expression, not just linear factors.
A question is this type if and only if you are told that a polynomial gives equal remainders when divided by two different linear expressions, and must use this to find constants.
A question is this type if and only if you must prove that a given equation (often involving logarithms or other functions) can be rearranged into a specific polynomial equation.
Questions where you solve equations like p(e^x) = 0, p(2^y) = 0, or p(3^t) = 0 by first solving p(x) = 0, then solving the resulting exponential equations using logarithms.
A question is this type if and only if you must show a polynomial can be written in a specific form like (x+a)(bx+c)² or as a product with given structure.
A question is this type if and only if you must find constants in an identity of the form p(x) ≡ (x+a)(bx²+cx+d)+e by expanding and comparing coefficients.
Questions where you solve p((y+c)^r) = 0 for rational exponents r (like (y+3)^(1/2) or (y-2)^2) by first solving p(x) = 0, then solving algebraic equations involving powers and roots.
Questions where the polynomial contains unknown constants that must first be determined using given remainder conditions, before verifying a factor and factorising.
Questions where one unknown constant is found using a single factor condition, then factorise and prove root count by examining the discriminant of the quadratic factor.
A question is this type if and only if you must find constants using information about specific coefficients in an expansion or product, rather than factor/remainder conditions.
Given a polynomial with unknown constants where at least one condition involves the derivative being zero or having a specific factor, requiring differentiation before applying factor/remainder theorem.
No factor is given; you must first use the factor theorem to find a factor by testing values, then factorise and solve completely.
Questions involving two distinct polynomials f(x) and g(x) where conditions are given separately for each, or a common factor is shared between them.
Questions that verify a factor, factorise completely, then use the factorisation to simplify a rational expression or algebraic fraction.
Questions that verify a given factor, factorise the cubic, then solve an equation where x is replaced by a trigonometric or other function (e.g., f(cosec θ) = 0).
A question is this type if and only if you must use a factorised polynomial to find an area or evaluate an integral, after first factorising.
Questions where the final equation involves solving for exponential expressions (e^x, 2^y, 3^t, 5^y, etc.) after factorising the polynomial.
Questions that verify a factor, factorise completely, then use the factorisation to solve a related equation (often involving substitution like y² for x) or prove divisibility properties.
Questions where a constant in the polynomial is unknown and must be found using the given factor, then the polynomial is factorised completely.
A question is this type if and only if you must explicitly find both the quotient and remainder when dividing a polynomial by a linear or quadratic expression.
A question is this type if and only if you must solve p(x) > 0 or p(x) < 0 by first factorising the polynomial and analysing sign changes.
Questions where a linear expression is stated to be a factor of the polynomial, requiring the polynomial to equal zero at a specific value.
Questions that verify a factor, factorise completely, then sketch the curve or find/analyse turning points and other curve properties.
Questions where two unknown constants must be found using two conditions (one factor condition and one remainder/value condition), then factorise and prove root count.
Both conditions give remainders directly, leading to a straightforward system of two equations in two unknowns without any intermediate 'show that' step.
Questions where division by a linear expression gives a specified non-zero remainder, requiring the polynomial to equal that remainder value at a specific point.
Questions where a constant appears as part of a coefficient expression (e.g., p+3 or p-15) requiring algebraic manipulation before solving simultaneous equations.
Given one factor of a polynomial with two or more unknown constants, express one constant in terms of another or find relationships between them using the factor theorem.
A question is this type if and only if you must sketch y = p(x) after finding roots and possibly turning points of the polynomial.
Questions where you solve p(sin θ) = 0, p(cos θ) = 0, or p(tan θ) = 0 by first solving p(x) = 0, then solving the resulting trigonometric equations for θ in a given interval.