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Cambridge IGCSE0580

Algebraic manipulation

Mathematics 0580 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 brackets
  3. 3Factorising by common factors
  4. 4Two letters and three brackets
  5. 5Grouping and the difference of two squares
  6. 6Factorising quadratics and cubics
  7. 7Completing the square

1. Collecting like terms

Like terms have exactly the same letters with the same powers. Only like terms can be added or subtracted. x and x² are not like terms, and neither are ab and a.

Simplify means give the answer in its simplest form. Keep the sign in front of each term. The constant terms are like terms too.

When you subtract an expression, write it in brackets first. Then change every sign inside.

7a + 3b + 5a - 9b = 12a - 6b

2a² + 3ab - 1 + 5a² - 9ab + 4 = 7a² - 6ab + 3

Examiner insight

“Some tried to combine unlike terms even after a correct answer was seen.” “Others obtained the correct answer in their working then made a transcription error to the answer line which is not permitted in algebra questions.”

Worked example 14 marks

Krish has 6x + 4 marbles.

Jo has 9 - x marbles.

(a) Write down an expression for the total number of marbles that Krish and Jo have. Give your answer in its simplest form.

(b) Krish has more marbles than Jo. Write down an expression for how many more marbles Krish has than Jo. Give your answer in its simplest form.

  1. 1

    (a) total = (6x + 4) + (9 - x)

  2. 2

    x terms: 6x - x = 5x [1]

  3. 3

    numbers: 4 + 9 = 13, so 5x + 13 final answer [1]

  4. 4

    (b) difference = (6x + 4) - (9 -x) = 6x + 4 - 9 + x [1]

  5. 5

    = 7x - 5 final answer [1]

2. Expanding brackets

To expand, multiply every term in one bracket by every term in the other. A single bracket gives 3x(2x - 4y) = 6x² - 12xy.

For two brackets there are four products. Watch the signs: a negative times a positive is negative.

Then collect like terms and give the final answer in simplest form.

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

(2x + 1)(x - 4) = 2x² - 7x - 4

Examiner insight

“Arithmetic slips were sometimes made when expanding the brackets or collecting terms.”

Worked example 12 marks

Expand the brackets and simplify.

(2x + 5)(x + 3)

  1. 1

    (2x + 5)(x + 3) = 2x² + 6x + 5x + 15 [1]

  2. 2

    = 2x² + 11x + 15 final answer [1]

3. Factorising by common factors

Factorising is the reverse of expanding. Take out the highest common factor of the numbers and of the letters.

The question says factorise, so factorise fully: nothing common may be left in the bracket. Check by expanding.

For 9x² + 15xy the common factor is 3x, so the answer is 3x(3x + 5y).

ax + ay = a(x + y)

9x² + 15xy = 3x(3x + 5y)

Examiner insight

“The factorising was well done by those who understood the topic, although a number of candidates only gained one mark for an incomplete factorisation.” “A few candidates partially factorised the expression, taking out a factor such as 3.”

Worked example 12 marks

Factorise.

9x³ - 6x

  1. 1

    common factor of 9 and 6 is 3; common letter is x, so take out 3x [1]

  2. 2

    9x³ - 6x = 3x(3x² - 2) final answer [1]

4. Two letters and three brackets

Two brackets can hold two letters. There are still four products, and the two xy terms are like terms.

For three brackets, expand two brackets first and collect like terms to a three-term expression. Multiply that by the third bracket, term by term.

Write out every product before you simplify. Then collect like terms for the final answer.

(3x + y)(x - 4y) = 3x² - 11xy - 4y²

(x + 4)(x - 2) = x² + 2x - 8

Examiner insight

“The most successful method was to multiply two brackets out to give a three-term quadratic expression and then multiply by the third bracket.” “Some candidates omitted terms when multiplying their quadratic expression (the result of multiplying two brackets) by the remaining bracket.”

Worked example 13 marks

Expand and simplify.

(3x + 1)(x + 4)(x - 2)

  1. 1

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

  2. 2

    (3x + 1)(x² + 2x - 8) = 3x³ + 6x² - 24x + x² + 2x - 8 [1]

  3. 3

    = 3x³ + 7x² - 22x - 8 final answer [1]

5. Grouping and the difference of two squares

Four terms: group them in pairs and take a common factor from each pair. If the same bracket appears twice, take it out as a common bracket. For ax + bx + kay + kby this gives (a + b)(x + ky).

Two squares with a minus between them: the difference of two squares factorises as (ax - by)(ax + by). Take out any number common to both terms first.

A perfect square a² + 2ab + b² is (a + b)².

ax + bx + kay + kby = (a + b)(x + ky)

a²x² - b²y² = (ax - by)(ax + by)

a² + 2ab + b² = (a + b)²

Examiner insight

“Those who did not score both marks often scored one mark for a correct partial factorisation with a repeated bracket and stopped at this stage.” “Many candidates were able to factorise out the 3 successfully, but from here failed to recognise the difference of two squares.”

Worked example 13 marks

Factorise fully.

18a² - 8b²

  1. 1

    take out the common factor: 18a² - 8b² = 2(9a² - 4b²) [1]

  2. 2

    9a² = (3a)² and 4b² = (2b)², a difference of two squares [1]

  3. 3

    = 2(3a + 2b)(3a - 2b) final answer [1]

6. Factorising quadratics and cubics

For ax² + bx + c, find two numbers with product ac and sum b. Split the middle term with them, then factorise in pairs.

Check by expanding. If the signs are reversed, the middle term comes out wrong.

For ax³ + bx² + cx, take out x first. Then factorise the quadratic that is left.

ax³ + bx² + cx = x(ax² + bx + c)

x² + 7x + 12 = (x + 3)(x + 4)

Examiner insight

“Strong candidates had no problem factorising the quadratic expression with little need to show any working although some did show splitting the 7x into 15x – 8x.” “Errors usually involved the signs reversed or going further than needed and treating the question as a quadratic equation.”

Worked example 13 marks

Factorise.

6x² + 7x - 20

  1. 1

    product 6 × (-20) = -120 and sum 7: the numbers are 15 and -8 [1]

  2. 2

    6x² + 15x - 8x - 20 = 3x(2x + 5) - 4(2x + 5) [1]

  3. 3

    = (2x + 5)(3x - 4) final answer [1]

7. Completing the square

Completing the square writes ax² + bx + c in the form a(x + p)² + q.

For x² + bx, halve the coefficient of x to get p. Then subtract the square p² to balance it.

If a is not 1, take a out of the x² and x terms first. Remember to multiply the subtracted square by a.

x² + bx + c = (x + b/2)² + c - b²/4

x² + 8x - 7 = (x + 4)² - 23

Examiner insight

“This question was one of the most challenging questions on the paper as it did not explicitly ask candidates to complete the square and many did not realise that this was what was expected of them.”

Worked example 13 marks

Write 3x² - 6x + 5 in the form a(x + b)² + c.

  1. 1

    take 3 out of the x² and x terms: 3x² - 6x + 5 = 3(x² - 2x) + 5 [1]

  2. 2

    half of -2 is -1: x² - 2x = (x - 1)² - 1 [1]

  3. 3

    3[(x - 1)² - 1] + 5 = 3(x - 1)² - 3 + 5 = 3(x - 1)² + 2 final answer [1]

Recap

  • Only like terms combine: same letters, same powers. Give the answer in simplest form.
  • Expanding: multiply every term by every term, then collect like terms.
  • Factorise fully: take out the highest common factor, then check by expanding.
  • Extended: two letters still give four products. Expand three brackets two at a time, writing every product.
  • Extended: four terms, group in pairs. A difference of two squares is (ax - by)(ax + by).
  • Extended: split the middle term for ax² + bx + c. Take out x first for a cubic.
  • Extended: complete the square as a(x + p)² + q. Halve the coefficient of x, then subtract p².

End-of-chapter exercise

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

  1. Do not use a calculator. Simplify 7a + 3b - 2a - 8b.2 marks
  2. Expand 5x(3x - 2y).2 marks
  3. Do not use a calculator. Expand and simplify 3(2x - 5) - 4(x - 2).3 marks
  4. Factorise fully 12ab - 18b².2 marks
  5. A rectangle has length (x + 7) cm and width (x - 2) cm. Write an expression for its area. Expand and simplify it.3 marks
  6. Do not use a calculator. Expand and simplify (x + 1)(x - 4)(2x + 3).3 marks
  7. Do not use a calculator. Factorise xy + 3y - 4x - 12.2 marks
  8. Factorise 3x³ + 7x² - 6x.3 marks
  9. A square has side (3x + 2) cm. A smaller square of side (x - 4) cm is cut from it. Write an expression for the area that is left. Give your answer fully factorised.4 marks
  10. Do not use a calculator. 5 + 12x - 2x² can be written in the form p - 2(x - q)². Find the value of p and the value of q.4 marks

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