x³Equation Station SA
Grade 12 CAPS · 20 marks · 20 minutes

Polynomials:
exam readiness

Original Paper 1-style practice across every core skill: sign handling, factors, cubics, parameters and an error check.

Start the paper
01
Original assessment

Polynomials mixed paper

Total: 20 marks. This is original practice, not an official past-paper question.

Before you startFor a Remainder or Factor Theorem question, first set the complete divisor equal to zero. That line prevents most sign and fraction errors.

Question 1 4 marks

Let \(A(x)=3x^3+5x^2-x+1\).

  1. Find the remainder when \(A(x)\) is divided by \(x+2\). 2
  2. Find the remainder when \(A(x)\) is divided by \(2x-1\). 2

Question 2 3 marks

\(B(x)=x^3+kx^2-5x-6\), where \(x-2\) is a factor.

  1. Find \(k\). 1
  2. Factorise \(B(x)\) fully. 2

Question 3 5 marks

Solve \(6x^3-5x^2-17x+6=0\). Show the factor that opens the cubic and state all roots.

Question 4 4 marks

\(C(x)=x^3+px^2-7x+q\). Given that \(x+1\) is a factor of \(C(x)\), and the remainder on division by \(x-2\) is \(-9\), determine \(p\) and \(q\).

Question 5 4 marks

A learner divides \(H(x)=x^3-4x+3\) by \(x-1\) using the coefficient row \(1,\,-4,\,3\).

  1. Explain the error in the coefficient row. 2
  2. Use the correct row to write the quotient. 2
Reveal the memo only after your complete attempt

Question 1

(a) \(x+2=0\Rightarrow x=-2\).

A(-2)=3(-8)+5(4)-(-2)+1=-24+20+2+1=-1

Remainder \(=-1\).

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

A(1/2)=3(1/8)+5(1/4)-1/2+1=17/8

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

Question 2

\(x-2\) factor \(\Rightarrow B(2)=0\).

8+4k-10-6=0 \Rightarrow 4k-8=0 \Rightarrow k=2
B(x)=x^3+2x^2-5x-6=(x-2)(x^2+4x+3)=(x-2)(x+1)(x+3)

Question 3

\(p(2)=48-20-34+6=0\), so \((x-2)\) is a factor.

6x^3-5x^2-17x+6=(x-2)(6x^2+7x-3)=(x-2)(3x-1)(2x+3)

\(x=2,\;\frac13,\;-\frac32\).

Question 4

Factor condition: \(C(-1)=0\Rightarrow p+q+6=0\).

Remainder condition: \(C(2)=-9\Rightarrow4p+q=-3\).

q=-p-6;\quad 4p-p-6=-3 \Rightarrow p=1,\ q=-7

Question 5

(a) The learner omitted the missing \(x^2\) term. The correct coefficient row is \(1,\,0,\,-4,\,3\).

(b) Synthetic division using root \(1\) gives bottom row \(1,\,1,\,-3,\,0\).

H(x)=(x-1)(x^2+x-3),\quad \text{so the quotient is }x^2+x-3.
02
Use your mark diagnostically

Choose the next repair

A score matters only if it tells you what to practise next.

16–20Ready for unfamiliar textbook and past-paper questions.
10–15Strong base; target the question types that cost marks.
0–9Return to the guided slides, then rebuild skill by skill.
Lost Question 1?Revisit the divisor-to-input table and practise Set B.
Lost Question 2 or 4?Revisit Factor Theorem and the two-conditions model.
Lost Question 3?Revisit candidate roots and completing the quadratic.
Lost Question 5?Revisit the synthetic division bridge and missing coefficients.
Best next move: repair one weak skill, then reattempt only that question before doing a new paper.Return to Mastery Bank
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Official scope and extra practice

This paper is original. Use the resources below for trusted curriculum detail and more questions.

Official scope

DBE CAPS Mathematics

Confirm the Grade 12 scope: cubics and theorem applications through degree 3.

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Extra exercises

Siyavula Polynomials

Use its Grade 12 sequence and end-of-chapter questions once you correct this paper.

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Official papers

DBE NSC archive

Use official past papers after you can complete the original practice independently.

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