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Exam questions · Maths · Algebra

Simplifying and Expanding Expressions

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

    Simplify \(7p + 2q - 3p - 5q\).

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    Model answer

    \(7p - 3p = 4p\) and \(2q - 5q = -3q\), so the answer is \(4p - 3q\).

    Mark scheme

    • \(4p\) or \(-3q\) correct — M1
    • \(4p - 3q\) — A1
  2. 2 Expand [3 marks]

    Expand and simplify \(3(2x + 1) + 4(x - 3)\).

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    Model answer

    \(3(2x + 1) = 6x + 3\) and \(4(x - 3) = 4x - 12\). Adding gives \(10x - 9\).

    Mark scheme

    • \(6x + 3\) or \(4x - 12\) correct — M1
    • \(6x + 3 + 4x - 12\) — M1
    • \(10x - 9\) — A1
  3. 3 Simplify [2 marks]

    Simplify (a) \(5b \times 2b^3\) [1 mark] (b) \(12m^5 \div 3m^2\) [1 mark]

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    Model answer

    (a) \(5 \times 2 = 10\) and \(b \times b^3 = b^4\), so \(10b^4\). (b) \(12 \div 3 = 4\) and \(m^5 \div m^2 = m^3\), so \(4m^3\).

    Mark scheme

    • (a) \(10b^4\) — B1
    • (b) \(4m^3\) — B1
  4. 4 Expand [2 marks]

    Expand and simplify \((x + 6)(x - 3)\).

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    Model answer

    \(x^2 - 3x + 6x - 18 = x^2 + 3x - 18\).

    Mark scheme

    • Three of the four terms correct, or \(x^2 - 3x + 6x - 18\) — M1
    • \(x^2 + 3x - 18\) — A1
  5. 5 Expand [3 marks]

    Expand and simplify \((3x - 1)(2x + 5)\).

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    Model answer

    \(6x^2 + 15x - 2x - 5 = 6x^2 + 13x - 5\).

    Mark scheme

    • At least three of the four terms correct — M1
    • \(6x^2 + 15x - 2x - 5\) — M1
    • \(6x^2 + 13x - 5\) — A1
  6. 6 Expand [2 marks]

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

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    Model answer

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

    Mark scheme

    • \((x - 4)(x - 4)\) with three or four terms correct — M1
    • \(x^2 - 8x + 16\) — A1
  7. 7 Show that [3 marks]

    Show that \((x + 5)^2 - 25 \equiv x(x + 10)\).

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    Model answer

    \((x + 5)^2 = x^2 + 10x + 25\). Subtracting 25 gives \(x^2 + 10x\), which is \(x(x + 10)\), as required.

    Mark scheme

    • \((x + 5)(x + 5)\) or \(x^2 + 5x + 5x + 25\) — M1
    • \(x^2 + 10x + 25 - 25\) or \(x^2 + 10x\) — M1
    • \(x(x + 10)\) with a conclusion — Q1

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: \((x^3)^2 = x^6\).