Finding a remainder without dividing, testing for factors, factorising cubics, and using the theorems to find unknown coefficients.
5 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.
“What is the remainder when this cubic is divided by x − 3?” You could divide — or you could just evaluate f(3). One number, one line.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
Test a value of x. If f(k) = 0, then (x − k) is a factor.
Copy these into your notebook in your own words — that is how they stick.
Remainder on division by (x − a) is f(a). By (ax − b): f(b/a).
(x − a) is a factor ⇔ f(a) = 0.
Try ±factors of the constant term (and ±factor ratios when the leading coefficient is not 1).
Root → divide → factorise the quotient.
Each condition gives one equation: factor → f(a) = 0; remainder r → f(a) = r.
Substitute the final coefficients back into f(a).
The remainder when f(x) is divided by a quadratic is linear: Ax + B.
f(a) means replace every x by a and evaluate.
3 quick games on this lesson. Earn XP and badges as you play — your best score for each game is saved. Games are for practice and do not change your exam readiness. More games in the Additional Mathematics Arcade.
8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.
Write full answers, then reveal the mark scheme and tick the points you earned (24 marks in total). Show all your working — method marks count. Your answers are saved on this device.
Question 1 [13 marks]
f(x) = x³ + ax² + bx + 6. When f(x) is divided by x − 2 the remainder is 12 and x − 1 is a factor.
(a) Show that a + b = −7 and 2a + b = −1. [4]
(b) Hence find a and b. [2]
(c) Factorise f(x) completely. [4]
(d) Solve f(x) = 0 correct to 2 decimal places. [3]
Question 2 [11 marks]
p(x) = 2x³ − 5x² + kx + 6 has x − 2 as a factor.
(a) Show that k = −1. [2]
(b) Factorise p(x) completely. [4]
(c) Find the remainder when p(x) is divided by x + 2. [2]
(d) Hence, or otherwise, solve p(x) > 0. [3]
Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.