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Exam questions · Maths · Algebra

Simplifying and Expanding Expressions

  • 7 exam questions
  • 19 marks
  • 10 quick checks
  1. 1 Simplify [2 marks]

    Simplify \(8a - 3b + 2a + 5b\).

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    Model answer

    \(8a + 2a = 10a\) and \(-3b + 5b = 2b\), so the answer is \(10a + 2b\).

    Mark scheme

    • \(10a\) or \(2b\) correct — M1
    • \(10a + 2b\) — A1
  2. 2 Expand [3 marks]

    Expand and simplify \(2(3x - 4) - 3(x - 5)\).

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    Model answer

    \(2(3x - 4) = 6x - 8\) and \(-3(x - 5) = -3x + 15\). So the total is \(6x - 8 - 3x + 15 = 3x + 7\).

    Mark scheme

    • \(6x - 8\) or \(-3x + 15\) correct — M1
    • \(6x - 8 - 3x + 15\) — M1
    • \(3x + 7\) — A1
  3. 3 Simplify [3 marks]

    (a) Simplify \(4x^2y \times 3xy^2\). [2 marks] (b) Expand \(x(x - 7)\). [1 mark]

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    Model answer

    (a) \(4 \times 3 = 12\), \(x^2 \times x = x^3\) and \(y \times y^2 = y^3\), so \(12x^3y^3\). (b) \(x \times x - x \times 7 = x^2 - 7x\).

    Mark scheme

    • (a) Two of \(12\), \(x^3\), \(y^3\) correct — M1
    • (a) \(12x^3y^3\) — A1
    • (b) \(x^2 - 7x\) — B1
  4. 4 Expand [2 marks]

    Expand and simplify \((x + 8)(x - 3)\).

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    Model answer

    \(x^2 - 3x + 8x - 24 = x^2 + 5x - 24\).

    Mark scheme

    • Three of the four terms correct, or \(x^2 - 3x + 8x - 24\) — M1
    • \(x^2 + 5x - 24\) — A1
  5. 5 Expand [3 marks]

    Expand and simplify \((4x - 3)(x + 2)\).

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    Model answer

    \(4x^2 + 8x - 3x - 6 = 4x^2 + 5x - 6\).

    Mark scheme

    • At least three of the four terms correct — M1
    • \(4x^2 + 8x - 3x - 6\) — M1
    • \(4x^2 + 5x - 6\) — A1
  6. 6 Show that [3 marks]

    Show that \((2x + 1)^2 - (2x - 1)^2 = 8x\).

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    Model answer

    \((2x + 1)^2 = 4x^2 + 4x + 1\) and \((2x - 1)^2 = 4x^2 - 4x + 1\). Subtracting: \(4x^2 + 4x + 1 - 4x^2 + 4x - 1 = 8x\), as required.

    Mark scheme

    • \((2x + 1)^2 = 4x^2 + 4x + 1\) — M1
    • \((2x - 1)^2 = 4x^2 - 4x + 1\) — M1
    • Subtracts correctly to reach \(8x\) — A1
  7. 7 Explain [3 marks]

    Dan says that \((x + 3)^2 = x^2 + 9\). (a) Explain what Dan has done wrong. [1 mark] (b) Expand and simplify \((x + 3)^2\). [2 marks]

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    Model answer

    (a) Dan has squared each term separately instead of multiplying the bracket by itself, so he has missed the middle term. (b) \((x + 3)(x + 3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9\).

    Mark scheme

    • (a) States that the middle term is missing, or that he has squared each term — B1
    • (b) \(x^2 + 3x + 3x + 9\) — M1
    • (b) \(x^2 + 6x + 9\) — A1

Quick check

  1. 1

    Expand and simplify \((x + 3)(x - 3)\).

    1. A\(x^2 - 6\)
    2. B\(x^2 + 9\)
    3. C\(x^2 - 6x - 9\)
    4. D\(x^2 - 9\)
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    D: \(x^2 - 9\)

    \(x^2 - 3x + 3x - 9 = x^2 - 9\), because the middle terms cancel.

  2. 2

    \(n\) is an integer. Which of these expressions is always an odd number?

    1. A\(2n + 1\)
    2. B\(n + 1\)
    3. C\(2n\)
    4. D\(2n + 2\)
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    A: \(2n + 1\)

    \(2n\) is always even, so \(2n + 1\) is always odd.

  3. 3

    Simplify \(4x + 3y - x + 2y\).

    1. A\(3x + 5y\)
    2. B\(5x + 5y\)
    3. C\(3x + y\)
    4. D\(8xy\)
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    A: \(3x + 5y\)

    Collect the \(x\) terms: \(4x - x = 3x\). Collect the \(y\) terms: \(3y + 2y = 5y\).

  4. 4

    Simplify \(3x \times 4x^2\).

    1. A\(12x^3\)
    2. B\(12x^2\)
    3. C\(7x^3\)
    4. D\(7x^2\)
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    A: \(12x^3\)

    Multiply the numbers (\(3 \times 4 = 12\)) and add the powers (\(x^1 \times x^2 = x^3\)).

  5. 5

    Expand \(3(2x - 5)\).

    1. A\(6x - 5\)
    2. B\(6x + 15\)
    3. C\(5x - 15\)
    4. D\(6x - 15\)
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    D: \(6x - 15\)

    Multiply both terms inside by 3: \(3 \times 2x = 6x\) and \(3 \times (-5) = -15\).

  6. 6

    Expand \(-2(x - 4)\).

    1. A\(2x - 8\)
    2. B\(-2x + 4\)
    3. C\(-2x - 8\)
    4. D\(-2x + 8\)
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    D: \(-2x + 8\)

    \(-2 \times x = -2x\) and \(-2 \times (-4) = +8\), so the answer is \(-2x + 8\).

  7. 7

    Expand and simplify \((x + 3)(x + 5)\).

    1. A\(x^2 + 8x + 8\)
    2. B\(2x + 8\)
    3. C\(x^2 + 15\)
    4. D\(x^2 + 8x + 15\)
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    D: \(x^2 + 8x + 15\)

    \(x^2 + 5x + 3x + 15 = x^2 + 8x + 15\).

  8. 8

    Expand and simplify \((x - 4)^2\).

    1. A\(x^2 - 8x - 16\)
    2. B\(x^2 + 16\)
    3. C\(x^2 - 8x + 16\)
    4. D\(x^2 - 16\)
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    C: \(x^2 - 8x + 16\)

    \((x - 4)(x - 4) = x^2 - 4x - 4x + 16 = x^2 - 8x + 16\).

  9. 9

    Which of these is an identity, true for every value of \(x\)?

    1. A\(x^2 = 4\)
    2. B\(3x + 1 = 10\)
    3. C\(2(x + 3) \equiv 2x + 6\)
    4. D\(x + 5 = 2x\)
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    C: \(2(x + 3) \equiv 2x + 6\)

    Expanding the bracket shows \(2(x + 3)\) is always equal to \(2x + 6\). The others are true only for certain values.

  10. 10

    Simplify \((2x^3)^2\).

    1. A\(4x^5\)
    2. B\(8x^6\)
    3. C\(4x^6\)
    4. D\(2x^6\)
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    C: \(4x^6\)

    Square the 2 to get 4 and multiply the powers: \((x^3)^2 = x^6\).