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Chapter 3 Practice Paper: Polynomials and Partial Fractions — question paper

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QQ/S3AM/CH3

Secondary 3 · Additional Mathematics (G3)

Chapter 3 Practice Paper

Polynomials and Partial Fractions

Duration: 40 minutes
Total: 30 marks
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Class:
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Date:

READ THESE INSTRUCTIONS FIRST

  • Write your name, class and register number in the spaces at the top of this page.
  • Answer all questions.
  • Show all necessary working. Omission of essential working will result in loss of marks.
  • Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.
  • The number of marks is given in brackets [ ] at the end of each question or part question.
  • The total number of marks for each paper is shown on its cover.

Original practice material written for the MOE Additional Mathematics syllabus. It is not an official MOE or school examination paper. Chapter 3 (A4 Polynomials and partial fractions).

For Examiner’s Use
Section A/ 30
Total/ 30

Section A: Structured questions[30 marks]

Answer all questions in the spaces provided. Show all necessary working — omission of essential working will result in loss of marks. Give exact answers unless told otherwise.

  1. 1

    The polynomial P is given by P(x) = 2x³ − 5x² + 4x − 1.

    1. (a)

      State the remainder when P(x) is divided by x − 2, with a reason for your method.

      [2]
    2. (b)

      It is given that P(x) = (x − 2)(2x² − x + 2) + 3. Verify this identity by expanding.

      [3]
  2. 2

    Solve the equation x³ − 7x + 6 = 0.

    1. (a)

      Show that x = 1 is a root.

      [1]
    2. (b)

      Factorise x³ − 7x + 6 completely.

      [3]
    3. (c)

      State the three roots of the equation.

      roots =[2]
  3. 3

    Express (9x + 9)/((x − 1)(x + 2)²) as a sum of partial fractions.

    1. (a)

      Write down the form of the decomposition, with unknown constants.

      [1]
    2. (b)

      Find A and C by substitution.

      [2]
    3. (c)

      Find B by comparing coefficients, and write the final decomposition.

      [3]
  4. 4

    This question is about dividing x³ + 2x² − 5 by x − 2.

    1. (a)

      Find the quotient and the remainder.

      [3]
    2. (b)

      Write x³ + 2x² − 5 in the form (x − 2)Q(x) + R.

      [1]
    3. (c)

      Use the remainder theorem to confirm the remainder, showing your substitution.

      [2]
  5. 5

    The polynomial f is given by f(x) = x³ + ax² + bx − 2, where a and b are constants. When f(x) is divided by x + 1 the remainder is 0, and when f(x) is divided by x − 2 the remainder is 12.

    1. (a)

      Write down the two equations linking a and b.

      [2]
    2. (b)

      Find the values of a and b.

      a, b =[2]
    3. (c)

      Factorise f(x) completely and hence solve f(x) = 0.

      [3]

— End of paper —