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Cambridge IGCSE0607

Algebraic manipulation

International Mathematics 0607 Chapter Notes

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Algebraic manipulation
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Algebraic manipulation notes

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In this chapter

  1. 1Collecting like terms
  2. 2Expanding a single bracket
  3. 3Expanding two brackets
  4. 4Squares of brackets and three brackets
  5. 5Factorising by a common factor
  6. 6Factorising by grouping
  7. 7Difference of two squares and perfect squares
  8. 8Factorising quadratics and cubics

1. Collecting like terms

To simplify is to give an answer in its simplest form. You do this by collecting like terms.

Like terms have the same letters with the same powers. Only like terms can be added or subtracted. Unlike terms stay separate.

In 2a² + 3ab - 1 + 5a² - 9ab + 4, the a² terms are like terms and the ab terms are like terms. a² and ab are not like terms. The answer is 7a² - 6ab + 3.

Examiner insight

“Some got to the correct simplified expression and then went on to incorrectly combine the unlike terms.” “Some treated the expression as if the second minus sign affected effected every number to its right instead of just the 2m.”

Worked example 12 marks

Simplify.

5m - 2 - 2m + 3

  1. 1

    m terms: 5m - 2m = 3m [1]

  2. 2

    numbers: -2 + 3 = 1, so the answer is 3m + 1 [1]

2. Expanding a single bracket

To expand a bracket, multiply every term inside it by the term outside.

3x(2x - 4y) = 6x² - 12xy. Multiply the numbers, then the letters, and keep the signs.

A bracket with a number in front of it, such as 2(3x - 1) + 5(2x + 5), is expanded first. Then collect the like terms to give 16x + 23.

a(b + c) = ab + ac

Examiner insight

“Some candidates multiplied the first term correctly and not the second term.” “Some candidates found the correct answer and then continued incorrectly to try and simplify it.”

Worked example 12 marks

Expand.

3x(5x² - 2)

  1. 1

    3x × 5x² = 15x³ [1]

  2. 2

    3x × (-2) = -6x, so the answer is 15x³ - 6x [1]

3. Expanding two brackets

With two brackets, multiply each term in the first bracket by each term in the second. You get four terms. Then collect the like terms to leave three.

(2x + 1)(x - 4) = 2x² - 8x + x - 4 = 2x² - 7x - 4.

Stop when the expression is expanded and simplified. Do not factorise it again.

(a + b)(c + d) = ac + ad + bc + bd

Examiner insight

“It was common for candidates to know how to multiply out the brackets and that they should get four terms.” “These were not always collected correctly to a three term answer.”

Worked example 12 marks

Expand and simplify.

(2x - 3)(x + 5)

  1. 1

    2x² + 10x - 3x - 15 [1]

  2. 2

    = 2x² + 7x - 15 [1]

4. Squares of brackets and three brackets

A bracket squared is the bracket written twice: (3x + 4)² = (3x + 4)(3x + 4) = 9x² + 24x + 16. Do not square each term separately.

For three brackets, expand one pair first. Then multiply the result by the third bracket. Write all the working down.

Take care with the signs in the first expansion. An error there makes every later line wrong.

Examiner insight

“There were many correct responses with candidates choosing to expand one pair of brackets and then multiply the result by the third bracket, with all working written down clearly.” “Some candidates tried a different method, multiplying terms from the 3 brackets in pairs. Almost inevitably, terms were omitted leading to an incorrect answer.”

Worked example 13 marks

Expand and simplify.

(2x + 3)(x - 1)(x + 5)

  1. 1

    (x - 1)(x + 5) = x² + 4x - 5 [1]

  2. 2

    (2x + 3)(x² + 4x - 5) = 2x³ + 8x² - 10x + 3x² + 12x - 15 [1]

  3. 3

    = 2x³ + 11x² + 2x - 15 [1]

5. Factorising by a common factor

To factorise is to put a common factor outside a bracket. Factorise fully: take out the largest number and every letter that divides all the terms.

9x² + 15xy = 3x(3x + 5y). Here 3 is the largest number that divides 9 and 15, and x is common to both terms.

In 8x + 12 the largest number is 4, so the answer is 4(2x + 3). The 4 stays outside the bracket. Where a whole term is taken out, 1 is left in the bracket, as in x - 5x² = x(1 - 5x).

Examiner insight

“Some candidates only took out 2 as a common factor.” “Some candidates divided both parts by 4 and wrote 2x + 3 for their answer.”

Worked example 12 marks

Factorise 12b⁴ - 3b.

  1. 1

    the largest number that divides 12 and 3 is 3, and b is common, so take out 3b [1]

  2. 2

    12b⁴ - 3b = 3b(4b³ - 1) [1]

6. Factorising by grouping

An expression with four terms, such as ax + bx + kay + kby, is factorised in pairs: x(a +b) + ky(a + b).

The two pairs share a common bracket. Take it out: (a + b)(x + ky).

When the second pair starts with a minus sign, take the minus sign out too: -bx + ab = -b(x - a). The sign inside the bracket changes. Check by multiplying out the brackets.

Examiner insight

“Whilst many candidates did approach this question correctly, many candidates then made small slips, often with the negative sign outside of a bracket, which they would have found if they had checked their answers by multiplying out their brackets.”

Worked example 12 marks

Factorise 2xy - 2ay - bx + ab.

  1. 1

    factorise in pairs: 2y(x -a) - b(x -a) [1]

  2. 2

    common bracket: (x - a)(2y -b) [1]

7. Difference of two squares and perfect squares

The difference of two squares has two square terms with a minus sign between them: a²x² - b²y² = (ax - by)(ax + by). The brackets have opposite signs, so there is no middle term.

Take out any common factor first: 18p² - 2 = 2(9p² - 1) = 2(3p - 1)(3p + 1).

A perfect square has the form a² + 2ab + b² = (a + b)². The middle term is twice the product of a and b, as in x² + 10x + 25 = (x + 5)².

Examiner insight

“Many candidates did not recognise the difference of two squares in the denominator.”

Worked example 12 marks

Factorise 16x² - 9y².

  1. 1

    16x² = (4x)² and 9y² = (3y)² [1]

  2. 2

    16x² - 9y² = (4x - 3y)(4x + 3y) [1]

8. Factorising quadratics and cubics

To factorise ax² + bx + c, find two numbers that multiply to ac and add to b. Use them to split the middle term, then factorise in pairs.

For ax³ + bx² + cx, take out the common factor x first. Then factorise the quadratic that is left: 2x³ + 7x² + 3x = x(2x² + 7x + 3) = x(2x + 1)(x + 3).

The answer is a product of brackets with whole numbers in them. Check it by expanding.

Examiner insight

“Some candidates attempted to use the quadratic formula to find the solutions and put these into the brackets, but they usually gave ( x − 0.8 )( x − 2 ) which did not score.”

Worked example 13 marks

Factorise.

4x² - 5x - 6

  1. 1

    numbers that multiply to -24 and add to -5 are -8 and 3 [1]

  2. 2

    4x² - 8x + 3x - 6 = 4x(x - 2) + 3(x - 2) [1]

  3. 3

    = (4x + 3)(x - 2) [1]

Recap

  • Simplify by collecting like terms. Only like terms can be added or subtracted.
  • Expand: multiply every term in the bracket by the term outside.
  • Two brackets: each term in the first times each term in the second, then collect.
  • Factorise fully: take out the largest number and every common letter. Check by expanding.
  • Extended: (a + b)² has four terms, not two. Three brackets: expand a pair first.
  • Extended: pairs give a common bracket; watch a minus sign outside a bracket. Difference of two squares has opposite signs.
  • Extended: for ax² + bx + c find two numbers that multiply to ac and add to b.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. (a) Simplify 6p - 4q - 2p + 9q. (b) Factorise 10p - 15.3 marks
  2. (a) Simplify 9y - 3 - 4y + 8. (b) Factorise your answer to part (a).3 marks
  3. Do not use a calculator. (a) Expand 4x(2x - 5). (b) Factorise fully 12x²y - 18xy².4 marks
  4. (a) Expand and simplify (x + 6)(x - 4). (b) Factorise fully 12x - 18.4 marks
  5. A rectangle has length (x + 5) cm and width (x + 2) cm. (a) Write the area as an expanded expression. (b) Write the perimeter in its simplest form. (c) Factorise your answer to part (b).5 marks
  6. Do not use a calculator. (a) Expand and simplify (5x - 2)². (b) Factorise x² + 14x + 49.4 marks
  7. Expand and simplify (x + 1)(x - 3)(x + 4).3 marks
  8. Do not use a calculator. (a) Factorise 6ax - 3bx - 2ay + by. (b) Factorise 49x² - 4y².4 marks
  9. Do not use a calculator. A square tile has sides of length (5x + 3) cm. A square piece with sides of length (x + 7) cm is cut out of the tile. (a) Show that the area of tile left is (24x² + 16x - 40) cm². (b) Factorise 24x² + 16x - 40 fully.6 marks
  10. Cuboid A has length (2x + 5) cm, width (2x - 5) cm and height 3x cm. Cuboid B has volume (12x³ - 60x² + 75x) cm³. (a) Expand and simplify an expression for the volume of cuboid A. (b) Factorise 12x³ - 60x² + 75x fully. (c) Find the volume of cuboid A minus the volume of cuboid B. Give your answer fully factorised.8 marks

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