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AS Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Algebraic division and the factor theorem

Dividing by a linear factor, the factor theorem, factorising cubics, finding unknown coefficients.

12 original questions, 63 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

f(x) = 2x3 + x2 − 13x + 6. Use the factor theorem to show that (x + 3) is a factor of f(x).

2 marks
Show answer
f(-3) = 0, so (x + 3) is a factor
2

Given that (x + 1) is a factor of x3 + 2x2 − 5x − 6

  1. divide x3 + 2x2 − 5x − 6 by (x + 1)
  2. hence factorise x3 + 2x2 − 5x − 6 completely.
4 marks
Show answer
a) x2 + x - 6 b) (x + 1)(x + 3)(x - 2)
3

g(x) = x3 − 4x2 + kx + 10, where k is a constant. Given that (x + 2) is a factor of g(x), find the value of k. You must make your method clear.

3 marks
Show answer
k = -7
4

f(x) = 2x3 − 5x2 − 4x + 3

  1. Show that (x + 1) is a factor of f(x).
  2. Hence factorise f(x) completely.
  3. Write down all the solutions of f(x) = 0
5 marks
Show answer
a) f(-1) = 0 b) (x + 1)(2x - 1)(x - 3) c) x = -1, 1/2, 3
5

Given that x3 − 7x − 6 ≡ (x + 1)(x2 + px + q), where p and q are constants,

  1. find the values of p and q
  2. hence factorise x3 − 7x − 6 completely.
4 marks
Show answer
a) p = -1, q = -6 b) (x + 1)(x - 3)(x + 2)
6

f(x) = x3 − 2x2 − 5x − 12

  1. Use the factor theorem to show that (x − 4) is a factor of f(x).
  2. Write f(x) in the form (x − 4)(x2 + bx + c), where b and c are integers.
  3. Hence show that the equation f(x) = 0 has exactly one real root.
5 marks
Show answer
a) f(4) = 0 b) (x - 4)(x2 + 2x + 3) c) x2 + 2x + 3 has discriminant -8 < 0, so x = 4 is the only real root
7

f(x) = x3 + ax2 + bx − 12, where a and b are constants. Given that (x − 3) and (x + 1) are both factors of f(x),

  1. find the values of a and b
  2. factorise f(x) completely.
6 marks
Show answer
a) a = 2, b = -11 b) (x - 3)(x + 1)(x + 4)
8

f(x) = x3 − x2 − 8x + 12

  1. Show that (x − 2) is a factor of f(x).
  2. Factorise f(x) completely.
  3. Sketch the curve with equation y = f(x), showing the coordinates of the points where the curve meets the coordinate axes.
xy
6 marks
Show answer
a) f(2) = 0 b) (x - 2)2(x + 3) c) touches at (2, 0), crosses at (-3, 0) and (0, 12)
9

g(x) = 3x3 − 7x2 + 2x + 2a, where a is a positive constant.

Given that (x − a) is a factor of g(x),

  1. show that a(3a − 4)(a − 1) = 0
  2. For the integer value of a, solve g(x) = 0, giving your answers in exact form.
7 marks
Show answer
a) shown b) x = 1, x = (2 ± √10)/3
10

The curve C has equation y = f(x), where f(x) = x3 + px2 + qx + 6 and p and q are constants. Given that (x + 2) is a factor of f(x) and that C passes through the point (2, −4),

  1. find the values of p and q
  2. hence find the coordinates of the points where C meets the x-axis.
7 marks
Show answer
a) p = -2, q = -5 b) (-2, 0), (1, 0), (3, 0)
11

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

  1. Show that the equation 8x − 7(4x) + 14(2x) − 8 = 0 can be written as u3 − 7u2 + 14u − 8 = 0, where u = 2x
  2. Hence solve 8x − 7(4x) + 14(2x) − 8 = 0
6 marks
Show answer
a) shown b) x = 0, 1, 2
12

A joiner in Glasgow makes a wooden storage box in the shape of a closed cuboid. The box has length (x + 4) cm, width x cm and height (x − 1) cm. The volume of the box is 96 cm3.

  1. Show that x3 + 3x2 − 4x − 96 = 0
  2. Use the factor theorem to show that (x − 4) is a factor of x3 + 3x2 − 4x − 96
  3. Hence show that x = 4 is the only real solution of the equation in part (a).
  4. Find the total surface area of the box.
8 marks
Show answer
a) shown b) f(4) = 0 c) quotient x2 + 7x + 24 has discriminant -47 < 0 d) 136 cm2

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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