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A4aExpanding brackets

Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Algebra

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Revision notes

Multiplying out brackets: a single term over a bracket and the product of two binomials (both tiers), and three binomials (Higher). It appears as 1 to 3 mark questions and inside equations, proofs and area problems.

Grade by grade

What you need to be able to do, from the first marks up to the top grade.

  1. 3
    Expand a single bracketMultiply every term inside by the term outside, e.g. \(4(2x - 3) = 8x - 12\).
  2. 4
    Expand and simplify two single bracketsExpand each bracket, then collect like terms, e.g. \(3(x + 2) - 2(x - 5) = x + 16\).
  3. 4
    Expand with a letter outside the bracketMultiply letters as well as numbers, e.g. \(2a(3a + b) = 6a^2 + 2ab\).
  4. 5
    Expand the product of two binomialsMultiply each term in the first bracket by each term in the second, e.g. \((x + 3)(x - 7) = x^2 - 4x - 21\).
  5. 5
    Expand a squared bracketWrite it out twice: \((x - 4)^2 = (x - 4)(x - 4) = x^2 - 8x + 16\).

Notes

Single brackets

  • Multiply the term outside by every term inside: \(5(2x - 3) = 10x - 15\).
  • A negative outside changes the sign of every term inside: \(-3(x - 4) = -3x + 12\).
  • Letters outside multiply too: \(2y(y + 5) = 2y^2 + 10y\).
  • 'Expand and simplify' means multiply out, then collect like terms: \(4(x + 3) - 2(x - 1) = 4x + 12 - 2x + 2 = 2x + 14\).

Two brackets

  • Every term in the first bracket multiplies every term in the second, so two binomials give 4 products.
  • Use a grid, or FOIL (First, Outer, Inner, Last): \((x + 3)(x - 7) = x^2 - 7x + 3x - 21 = x^2 - 4x - 21\).
  • For a squared bracket, write the bracket twice: \((x + 5)^2 = (x + 5)(x + 5) = x^2 + 10x + 25\), never \(x^2 + 25\).
  • With coefficients: \((2x - 3)(x + 4) = 2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12\).
  • Difference of two squares: \((a + b)(a - b) = a^2 - b^2\), because the middle terms cancel.

Cheatsheet

  • \(a(b + c) = ab + ac\)
  • \((a + b)(c + d) = ac + ad + bc + bd\)
  • \((x + a)^2 = x^2 + 2ax + a^2\)
  • \((x - a)^2 = x^2 - 2ax + a^2\)
  • \((x + a)(x - a) = x^2 - a^2\)
  • \((x + a)(x + b) = x^2 + (a + b)x + ab\)

How to answer each type of question

Expand and simplify two single brackets

2 marks4
  1. Multiply out each bracket, taking care with the sign in front of the second one.
  2. Collect the letter terms and the number terms.

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

Show the model answer
\(6y - 15 - 4y + 8\) M1 (at least 3 terms correct)
\(2y - 7\) A1

Expand and simplify two binomials

2 marks5
  1. Write down all four products (grid or FOIL).
  2. Collect the two x terms.

Example. Expand and simplify \((x - 6)(x + 2)\)

Show the model answer
\(x^2 + 2x - 6x - 12\) M1 (at least 3 of the 4 terms correct)
\(x^2 - 4x - 12\) A1

Shortcuts and memory tricks

  • Check by substituting \(x = 1\) into the question and your answer: \((1 - 6)(1 + 2) = -15\) and \(1 - 4 - 12 = -15\), so they match.
  • For \((x + a)(x + b)\): the x term is \(a + b\) and the number term is \(a \times b\).
  • Squared bracket: square the first term, double the product of the two terms, square the last term.
  • Count your products: two binomials give 4 products before simplifying.

Where marks are lost

  • Writing \((x + 3)^2 = x^2 + 9\): the middle term \(6x\) is missing.
  • Only multiplying the first term: \(3(x + 4) = 3x + 4\) is wrong; it is \(3x + 12\).
  • Sign errors with a negative outside: \(-2(x - 5)\) is \(-2x + 10\), not \(-2x - 10\).
  • Stopping before collecting like terms when the question says 'simplify'.

Exam technique

  • 'Expand' means multiply out; 'expand and simplify' also means collect like terms.
  • Show all the products before simplifying so the method mark can be seen.
  • In 'show that' questions, write the final line so it matches the given expression exactly.

Sample questions

Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.

Question 1Easy3 marks
Jake says, '\((x - 5)^2 = x^2 - 25\) for all values of \(x\).'
(a) By substituting a value of \(x\), show that Jake is wrong.[1]
(b) Expand and simplify \((x - 5)^2\).[2]
Show the answer and mark scheme
(a) Answer: E.g. \(x = 1\): \((1 - 5)^2 = 16\) but \(1^2 - 25 = -24\).
  • C1 for a correct substitution showing the two sides are different, e.g. \(x = 1\) gives 16 and −24

Worked solution: When \(x = 1\): \((x - 5)^2 = (-4)^2 = 16\) and \(x^2 - 25 = -24\). These are different, so Jake is wrong.

(b) Answer: \(x^2 - 10x + 25\)
  • M1 for \(x^2 - 5x - 5x + 25\) (four terms, at least three correct)
  • A1 for \(x^2 - 10x + 25\)

Worked solution: \((x - 5)(x - 5) = x^2 - 5x - 5x + 25 = x^2 - 10x + 25\)

Question 2Medium4 marks
(a) Expand and simplify \(4m(4m - n) - 2n(m - n)\)[2]
(b) Expand \(6w(4w^{4} - 6)\)[2]
Show the answer and mark scheme
(a) Answer: \(16m^2 - 6mn + 2n^2\)
  • M1 for at least three of the four terms in \(16m^2 - 4mn - 2mn + 2n^2\) correct
  • A1 for \(16m^2 - 6mn + 2n^2\) cao

Worked solution: \(4m(4m - n) - 2n(m - n) = 16m^2 - 4mn - 2mn + 2n^2 = 16m^2 - 6mn + 2n^2\)

(b) Answer: \(24w^{5} - 36w\)
  • B2 for \(24w^{5} - 36w\) (B1 for one correct term)

Worked solution: Multiply each term in the bracket by \(6w\) and add the powers of \(w\): \(6w(4w^{4} - 6) = 24w^{5} - 36w\)

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