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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 \(5a + 3b - 2a + 4b\).

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    Model answer

    Collect the \(a\) terms: \(5a - 2a = 3a\). Collect the \(b\) terms: \(3b + 4b = 7b\). The answer is \(3a + 7b\).

    Mark scheme

    • \(3a\) or \(7b\) correct — M1
    • \(3a + 7b\) — A1
  2. 2 Expand [3 marks]

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

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    Model answer

    \(4(x + 3) = 4x + 12\) and \(-2(x - 1) = -2x + 2\). So the total is \(4x + 12 - 2x + 2 = 2x + 14\).

    Mark scheme

    • \(4x + 12\) or \(-2x + 2\) correct — M1
    • \(4x + 12 - 2x + 2\) — M1
    • \(2x + 14\) — A1
  3. 3 Simplify [3 marks]

    Simplify (a) \(3a \times 4a^2\) (1 mark) (b) \((2x^2y)^3\) (2 marks)

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    Model answer

    (a) Multiply the numbers and add the powers: \(3 \times 4 = 12\) and \(a \times a^2 = a^3\), so \(12a^3\). (b) Cube every part: \(2^3 \times (x^2)^3 \times y^3 = 8x^6y^3\).

    Mark scheme

    • (a) \(12a^3\) — B1
    • (b) Any two of \(8\), \(x^6\), \(y^3\) correct — M1
    • (b) \(8x^6y^3\) — A1
  4. 4 Expand [2 marks]

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

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    Model answer

    \(x^2 - 2x + 5x - 10 = x^2 + 3x - 10\).

    Mark scheme

    • Four correct terms, \(x^2 - 2x + 5x - 10\), or three of the four correct — M1
    • \(x^2 + 3x - 10\) — A1
  5. 5 Expand [3 marks]

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

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    Model answer

    \(2x \times x = 2x^2\), \(2x \times (-4) = -8x\), \(3 \times x = 3x\), \(3 \times (-4) = -12\). So the expansion is \(2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12\).

    Mark scheme

    • At least three of the four terms correct — M1
    • \(2x^2 - 8x + 3x - 12\) — M1
    • \(2x^2 - 5x - 12\) — A1
  6. 6 Show that [3 marks]

    A rectangle has length \((x + 3)\) cm and width \((2x - 1)\) cm. Show that the area of the rectangle is \((2x^2 + 5x - 3)\) cm\(^2\).

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    Model answer

    Area \(= (x + 3)(2x - 1) = 2x^2 - x + 6x - 3 = 2x^2 + 5x - 3\), as required.

    Mark scheme

    • Area \(= (x + 3)(2x - 1)\) — M1
    • At least three of the four terms \(2x^2\), \(-x\), \(6x\), \(-3\) correct — M1
    • \(2x^2 - x + 6x - 3\) leading to \(2x^2 + 5x - 3\) — C1
  7. 7 Explain [3 marks]

    Dan says that \((x + 3)^2 = x^2 + 9\). (a) Explain why Dan is wrong. (1 mark) (b) Expand and simplify \((x + 3)^2\). (2 marks)

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    Model answer

    (a) Dan has not included the middle term. Squaring a bracket means multiplying it by itself, so there are four terms, not two. (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 only squared each term — C1
    • (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: \((x^3)^2 = x^6\).