Grade 12 CAPS · 12 marks · 10 minutes

Polynomials:
test yourself

Work without notes. This compact test checks the full route: factor condition, factorisation, all roots, then a non-monic remainder.

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Paper 1 style

Polynomials mini-assessment

Total: 12 marks. Work on paper first, then reveal the marking guide below.

Before you startFor a theorem question, write the substitution value explicitly. A correct answer with no route is weaker than a clear exam method.

Question 1 10 marks

Let \(p(x)=x^3+kx^2+x+6\). Given that \(x-2\) is a factor of \(p(x)\):

Use the Factor Theorem to find \(k\). 2
Factorise \(p(x)\) fully. 3
Solve \(p(x)=0\). 3
Find the remainder when \(p(x)\) is divided by \(x+2\). 2

Question 2 2 marks

Find the remainder when \(q(x)=3x^3+5x^2-x+1\) is divided by \(2x-1\).

Reveal marking guide only after your full attempt

Question 1

(a) \(x-2\) factor means \(p(2)=0\).

8 + 4k + 2 + 6 = 0 ⇒ 4k + 16 = 0 ⇒ k = −4

(b) \(p(x)=x^3-4x^2+x+6\).

x³ − 4x² + x + 6 = (x−2)(x²−2x−3) = (x−2)(x−3)(x+1)

(c) \((x-2)(x-3)(x+1)=0\).

x = 2, 3, −1

(d) For \(x+2\), substitute \(x=-2\):

p(−2) = −8 − 16 − 2 + 6 = −20

Remainder = \(-20\).

Question 2

\(2x-1=0\Rightarrow x=\frac12\).

q(1/2) = 3(1/8) + 5(1/4) − 1/2 + 1 = 17/8

Remainder = \(\frac{17}{8}\).

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10–12Ready: do extra unfamiliar questions.

7–9Revisit the exact step you lost.

0–6Redo slides, then Sets B–D.

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This test is original. Use these sources for scope and additional textbook work.

Official scope

DBE CAPS Mathematics

Grade 12 Polynomials: cubic factorisation and theorem application through degree 3.

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Extra exercises

Siyavula Polynomials

Use its sequence for more questions after you correct the test.

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Repair practice

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