IP AMaths Notes (Upper Sec, Year 3-4): 09) Polynomials

Study guide08 Nov 2025, 00:00 ZUpdated 17 Jul 2026

Factor theorem, remainder theorem, and inequalities involving higher-degree polynomials for IP AMaths.

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For Integrated Programme students: Your current school materials, teacher instructions, and assessment scope take precedence because IP topic sequence and depth vary by school. This is an Eclat IP guide, not the O-Level / SEC G3 exam-track guide.

How this chapter applies

Eclat core: multiplication, division, remainder and factor theorems, cubic equations, and sum or difference of cubes form the main route.
School-sensitive extension: Vieta relations, cubic inequalities, and parameter-heavy factor problems may vary by school.
2027 national comparison: K341 Topic A4 covers polynomial multiplication and division, remainder and factor theorems, cubic equations, cube identities, and partial fractions.
Check your school: confirm whether synthetic division or Vieta methods are permitted or expected.
Exam-track route: use the separate O-Level and SEC G3 Additional Mathematics notes for K341 topic ownership.

Q: What does IP AMaths Notes (Upper Sec, Year 3-4): 09) Polynomials cover?
A: Factor theorem, remainder theorem, and inequalities involving higher-degree polynomials for IP AMaths.

Polynomials underpin factorisation, curve sketching, and inequality solving. Memorise the theorems that convert substitution into proofs of factor status.

Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.

New to the Integrated Programme? Start with What is IP? | Browse all free IP notes.

The core idea is simple: Polynomial questions turn substitution into factor and remainder information.

Use it as a working check: Use f(a) = 0 to prove x - a is a factor, then factor fully before solving inequalities or finding roots.

Then go one layer deeper: Example: if f(2) = 0, divide by x - 2 first. The remaining quadratic often unlocks the full factorisation and the sign chart.

Choosing the polynomial route

Read the wording before expanding. Polynomial questions usually tell you whether substitution, division, coefficient comparison, or a sign chart should come first.

Question cue

Reviewed by

Marcus Pang·Managing Director (Maths)

Known information	Set up	Compare coefficients	Result
$x - 2$ is a factor of $x^3 - 4x^2 + x + 6$	$x^3 - 4x^2 + x + 6 = (x - 2)(x^2 + px + q)$	Expand: $x^3 + (p - 2)x^2 + (q - 2p)x - 2q$	Match constants: $-2q = 6$ , so $q = -3$ .
Same setup	Use the $x^2$ coefficient.	$p - 2 = -4$	$p = -2$
Check the middle coefficient	Substitute $p = -2$ , $q = -3$ .	$q - 2p = -3 - 2(-2) = 1$

Factor pattern	Root behaviour	Sign-chart effect	Common trap
$(x - a)$ appears once	The graph crosses the axis at $x = a$ .	The sign changes across $a$ .	Forgetting to switch signs after a simple root.
$(x - a)^2$ appears	The graph touches the axis at $x = a$ .	The sign usually stays the same across $a$ .	Splitting the solution interval as if the sign changed.
$(x - a)^3$ appears	The graph crosses with flattening at $x = a$ .	The sign changes across $a$ .	Treating every repeated root as a no-change root.
Inequality uses $\leq$ or $\geq$	Roots that make the expression zero may be included.	Use square brackets or include the exact root where appropriate.	Including roots for strict $\lt$ or $\gt$ inequalities.

IP AMaths Notes (Upper Sec, Year 3-4): 09) Polynomials

How this chapter applies

Choosing the polynomial route

Sources

1 Key results

2 Worked example - Factorisation

Quotient recovery checkpoint

Sign-chart root behaviour checkpoint

3 Worked example - Polynomial inequality

4 Worked example - Using factor and remainder conditions

5 Practice Quiz

6 Try this

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