More about Polynomials
Master the Remainder and Factor Theorems. Learn how to factorize cubic polynomials step-by-step and discover how to write your steps to match the HKEA marking scheme perfectly.
The Core Concept
The Geometry of division: When you divide a polynomial f(x) by a linear divisor
ax − b, you get a quotient q(x) and a constant remainder R. We can write this relation as f(x) = (ax − b)q(x) + R. If the remainder R is exactly zero, then (ax − b) divides f(x) perfectly and is called a factor.DSE More about Polynomials Cheat Sheet — 5 Key Topics
Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.
| Topic | Key Concept & Steps | Exam Shortcuts & Tips |
|---|---|---|
| 1. Remainder Theorem | When f(x) is divided by ax − b, the remainder R is:R = f(b/a) | Watch your signs: • If divided by x − 2 ⟹ R = f(2)• If divided by 2x + 3 ⟹ R = f(−3/2) |
| 2. Factor Theorem | (ax − b) is a factor of f(x) if and only if:f(b/a) = 0 | If x − a is a factor, substitute x = a and set the entire expression equal to 0 to solve for unknown coefficients. |
| 3. Factorizing Cubics | To factorize Ax³ + Bx² + Cx + D:1. Guess a root r such that f(r) = 0 (try ±1, ±2).2. Divide f(x) by x − r to get a quadratic quotient.3. Factorize the quadratic term further. | DSE Marking Step: You must write down the division step (long division or synthetic division) to get the "M-mark" (Method mark) in Paper 1. |
| 4. HCF & LCM | 1. Factorize every polynomial completely first. • HCF: Product of the lowest powers of common factors. • LCM: Product of the highest powers of all factors. | Remember to factorize algebraic identities first, such as: • x² − y² = (x − y)(x + y)• x³ − y³ = (x − y)(x² + xy + y²) |
| 5. Solving Equations | To solve f(x) = 0:1. Factorize f(x) completely. 2. Set each factor to 0 and solve for x. | Check the quadratic factor's discriminant (Δ = b² − 4ac). If Δ < 0, that part yields no real roots. |
💡 Exam Traps & Secrets
⚠️ The Divisor Sign TrapMany students make a careless sign error when substituting x values. Rule: Always set the divisor to 0 and solve for x before substituting. Example: If f(x) is divided by
2x + 3, set 2x + 3 = 0 ⟹ x = −3/2. The remainder is f(−3/2), not f(3/2) or f(3).⚠️ "Factorize Completely" vs. "Solve the Equation" (DSE Marking Trap)In Paper 1 Section A, a question will often have two sub-parts:
(a) Factorize f(x) completely. ⟹ Your final answer must be a product of factors, e.g.,
(b) Hence, solve f(x) = 0. ⟹ Your final answer must be values of x, e.g.,
If you write the values of x in part (a), you will receive zero marks for the final accuracy step!
(a) Factorize f(x) completely. ⟹ Your final answer must be a product of factors, e.g.,
(x − 2)(x + 3)(2x − 1).(b) Hence, solve f(x) = 0. ⟹ Your final answer must be values of x, e.g.,
x = 2, x = −3, or x = 1/2.If you write the values of x in part (a), you will receive zero marks for the final accuracy step!
⚠️ The Quadratic Discriminant CheckWhen you factorize a cubic into a linear and a quadratic term, like
f(x) = (x − 2)(x² + 2x + 5), always check if the quadratic part can be factorized further. Calculate its discriminant: Δ = b² − 4ac = 2² − 4(1)(5) = −16 < 0. Since Δ < 0, x² + 2x + 5 cannot be factorized over real numbers. You must stop here; this is the final "completely factorized" answer.✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Remainder Theorem constant-solving)When
f(x) = x³ + kx² − 2x + 3 is divided by x − 2, the remainder is 7. Find the value of k.(Factor Theorem + Cubic Factorization)Let
f(x) = 2x³ + 5x² − x + k. Given that x + 3 is a factor of f(x), find the value of k and factorize f(x) completely.(Advanced HCF/LCM using algebraic identities)Find the HCF and LCM of
x² − 5x + 6, x² − 4, and x³ − 8.More about Polynomials Exercises: DSE-Style Practice
DSE-style practice sets from the SketchMath question bank, with solutions for every MC and long question.
Frequently Asked Questions
Dividing f(x) by (ax − b) gives remainder f(b/a). You never need to perform the full polynomial division — just substitute x = b/a into the polynomial. Be careful with signs: for divisor 2x + 3, set 2x + 3 = 0 so x = −3/2.
Try x = ±1, ±2, ±3, … until f(a) = 0. Then (x − a) is a factor. Divide f(x) by (x − a) using long division to get a quadratic, then factorize the quadratic. Always check the discriminant of the quadratic to see if further factorization is possible.
Section A asks you to find an unknown coefficient using Remainder/Factor Theorem and then factorize. The question often has two sub-parts: (a) factorize completely → product of factors; (b) hence solve f(x) = 0 → values of x. Mixing up these answer formats is a common marking trap.