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

Study guide08 Nov 2025, 00:00 Z

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

Download PDF Join our Telegram study group

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.

These tools align with the SEAB GCE O-Level Additional Mathematics (4049) syllabus: factor and remainder theorems, Vieta relationships, and sign-chart reasoning for cubic inequalities.

Status: SEAB O-Level Additional Mathematics 4049 syllabus (exams from 2025) checked 2025-11-30 - scope unchanged; remains the reference for this note.

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	Best first move	What to watch
"Show that $x - a$

Reviewed by

Marcus Pang·Managing Director (Maths)

Sources

SEAB GCE O-Level Additional Mathematics (4049) syllabus (examinations from 2025)

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

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

Related Posts