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IGCSE Mathematics 0580 · Chapter 3 notes · ~18 min read

Algebraic manipulation

Collect, expand, factorise, simplify and rearrange with confidence. Area models, FOIL arrows and step-by-step flow charts show why every algebra rule works, not just what to do.

ExpandingFactorisingAlgebraic fractionsRearranging formulaeSubstitution
RSby Riyaz Sir
mentorriyaz.in
No tag = Core (everyone)EXTENDED Extended onlyhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • Algebra is arithmetic with letters. Collect like terms, multiply every term when you expand, and always take out the common factor first when you factorise.
  • Expanding and factorising are opposites: (x + 2)(x + 3) ⟷ x² + 5x + 6. Rearranging a formula means undoing the operations in reverse order.
  • Extended: factorise quadratics, difference of two squares and grouping; simplify algebraic fractions; rearrange when the subject appears twice.

By the end you can

  • Simplify expressions by collecting like terms, and use the index laws with letters.
  • Write expressions and formulae from words, and substitute numbers (including negatives) into them.
  • Expand single and double brackets, and take out common factors.
  • Rearrange simple formulae to change the subject.
  • Factorise quadratics, the difference of two squares and by grouping. EXT
  • Simplify, add and subtract algebraic fractions. EXT
  • Rearrange formulae when the subject appears twice or inside a root. EXT
  • Use algebra to show that a statement is always true. EXT

1Algebra language

Terms and like terms

3x²−5xy+ 7expression = terms joined by + and −termcoefficient 3variable xtermcoefficient −5variables x and yconstant term(no letter)the sign belongsto the termafter itCollect like terms (same letters, same powers):4a + 3b − a + 2b= 3a + 5ba and b are different letters, so 3a + 5b cannot be simplified further
Every term has a sign in front of it. Only terms with exactly the same letters and powers are like terms.

Expression — letters and numbers joined by + and −, with no equals sign: 3x + 5.

Equation — two expressions joined by an equals sign that is true for certain values: 3x + 5 = 20.

Formula — a rule linking quantities, for example A = πr².

  • Like terms have identical letters and powers: 3x² and 5x² are like, but 3x² and 3x are not.
  • Multiply: numbers first, then letters, adding powers: 3x² × 4x³ = 12x⁵. Divide: subtract powers: 12x⁵ ÷ 4x² = 3x³.
  • Power of a power: (2x³)² = 2² × (x³)² = 4x⁶. Square the number as well as the letter. (More on the laws in Number toolkit.)
Worked example

Simplify 5a + 3b − 2a + 4b.

  1. collect the a terms: 5a − 2a = 3a
  2. collect the b terms: 3b + 4b = 7b
  3. answer: 3a + 7b
Worked example

Simplify 3x²y × 4xy³.

  1. numbers: 3 × 4 = 12
  2. x: x² × x = x³
  3. y: y × y³ = y⁴
  4. answer: 12x³y⁴

From words to algebra

5 more than xx + 53 less than xx − 3not 3 − xtwice x2xx squared, then doubled2x²x doubled, then squared(2x)²x shared between 4x ÷ 4
Read each word carefully. "Less than" reverses the order, and the order of "squared" and "doubled" matters.
Worked example

A pen costs £p and a book costs £b. Write an expression for the cost of 3 pens and 2 books.

  1. 3 pens = 3p
  2. 2 books = 2b
  3. total cost = 3p + 2b pounds
Worked example

A rectangle has sides (x + 3) and 2x. Write a simplified expression for its perimeter.

  1. perimeter = 2 × (x + 3) + 2 × 2x
  2. = 2x + 6 + 4x
  3. = 6x + 6

2Expanding brackets

Single brackets and FOIL

Single bracketDouble bracket: FOIL3(2x−5)3 × 2x + 3 × (−5)= 6x − 15Multiply the numberoutside by EVERY terminside the bracket.(x+3)(x−2)First: x × x = x²Outer: x × (−2) = −2xInner: 3 × x = +3xLast: 3 × (−2) = −6→ x² + x − 6
Every term inside the first bracket multiplies every term inside the second. Watch the signs.
  • Single bracket: multiply the outside by each term inside. 3(2x − 5) = 6x − 15.
  • Negative outside: the sign changes both terms. −2(x − 4) = −2x + 8.
  • Double brackets (FOIL): First, Outer, Inner, Last, then collect like terms. (x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6.
  • Squared bracket: write it twice. (x + 5)² = (x + 5)(x + 5) = x² + 10x + 25.
Common mistake(x + 5)² = x² + 25. No! You must write the bracket twice and expand. The middle term 10x is missing.
Worked example

Expand and simplify 3(x + 2) + 2(x − 1).

  1. 3(x + 2) = 3x + 6
  2. 2(x − 1) = 2x − 2
  3. add: 3x + 6 + 2x − 2
  4. = 5x + 4
Worked example

Expand and simplify (2x − 1)(x + 4).

  1. First: 2x × x = 2x²
  2. Outer: 2x × 4 = +8x
  3. Inner: −1 × x = −x
  4. Last: −1 × 4 = −4
  5. collect: 2x² + 8x − x − 4 = 2x² + 7x − 4
Worked example

Expand (3x − 2)².

  1. write twice: (3x − 2)(3x − 2)
  2. 9x² − 6x − 6x + 4
  3. = 9x² − 12x + 4

The area model

x²3x2x6x3x2(x + 3)(x + 2)= x² + 3x + 2x + 6= x² + 5x + 6the four areas are thefour productsCheck with x = 1: 4 × 3 = 12 and 1 + 5 + 6 = 12 ✓
The four areas inside the rectangle are exactly the four products in FOIL.
  • Draw a rectangle with the first bracket across the top and the second down the side. Multiply the label on the row by the label on the column in each cell, then add the cells.
  • Check: substitute a simple number (like x = 1) into the question and the answer. They must match.
Worked example

EXT Expand and simplify (x + 1)(x + 2)(x + 3).

  1. expand two brackets first: (x + 1)(x + 2) = x² + 3x + 2
  2. multiply by the third: (x² + 3x + 2)(x + 3)
  3. = x³ + 3x² + 3x² + 9x + 2x + 6
  4. collect: x³ + 6x² + 11x + 6
Memory trick"Smiley face": draw arcs from the first bracket's terms to the second bracket's terms. You should have 4 arcs for two terms each.

3Factorising

Factorising: the plan

1. Take out the highest common factor first6x² + 9x = 3x(2x + 3)2 termsdifference of two squaresa² − b²= (a − b)(a + b)x² − 49= (x − 7)(x + 7)3 termsfind two numbersmultiply to give cadd to give bx² + 5x + 6= (x + 2)(x + 3)4 termsgroup in pairsax + ay + bx + by= a(x + y) + b(x + y)x² + 3x + 2x + 6= (x + 2)(x + 3)
Always start by taking out the highest common factor, then count the terms.
  • Factorising puts the brackets back. It is the reverse of expanding. Always check by expanding your answer.
  • Highest common factor (HCF): the biggest number and the highest power of each letter that divides every term. 12x² − 18x: the HCF is 6x.
Worked example

Factorise 12x² − 18x.

  1. HCF of 12 and 18 is 6, and of x² and x is x
  2. HCF = 6x
  3. 12x² ÷ 6x = 2x and 18x ÷ 6x = 3
  4. answer: 6x(2x − 3)
Worked example

Factorise 4a²b + 6ab².

  1. HCF of 4 and 6 is 2
  2. a: lowest power is a, b: lowest power is b
  3. HCF = 2ab
  4. 4a²b ÷ 2ab = 2a, 6ab² ÷ 2ab = 3b
  5. answer: 2ab(2a + 3b)
Common mistakeFactorising 12x² − 18x as 2(6x² − 9x). This is not fully factorised. The bracket still has a common factor, so take out the highest one (6x).

Factorising quadraticsEXTENDED

2x²x6x32x1x31. a × c = 2 × 3 = 62. two numbers: product 6,sum 7 → they are 1 and 63. 2x² + x + 6x + 34. = x(2x + 1) + 3(2x + 1)5. = (2x + 1)(x + 3)the four cells add to 2x² + 7x + 3
When the number in front of x² is not 1, split the middle term using two numbers that multiply to a × c.
  • EXT x² + bx + c: find two numbers that multiply to c and add to b. These go in the brackets.
  • EXT ax² + bx + c: find two numbers that multiply to a × c and add to b, split the middle term, then factorise in pairs.
  • EXT Signs: if c is positive, both numbers have the same sign. If c is negative, the signs are different.
Worked example

EXT Factorise x² + 5x + 6.

  1. two numbers: multiply to 6, add to 5
  2. 2 × 3 = 6 and 2 + 3 = 5
  3. answer: (x + 2)(x + 3)
Worked example

EXT Factorise x² − 2x − 15.

  1. multiply to −15, add to −2
  2. different signs: −5 and +3
  3. answer: (x − 5)(x + 3)
Worked example

EXT Factorise 6x² + x − 2.

  1. a × c = 6 × (−2) = −12
  2. two numbers that multiply to −12 and add to +1: +4 and −3
  3. split: 6x² + 4x − 3x − 2
  4. pairs: 2x(3x + 2) − 1(3x + 2)
  5. answer: (3x + 2)(2x − 1)
Exam tipAlways expand your answer to check it. (3x + 2)(2x − 1) = 6x² − 3x + 4x − 2 = 6x² + x − 2 ✓

Difference of two squaresEXTENDED

aa12b²cut andmovea + ba − b12L-shape area: a² − b² = (a + b)(a − b)Piece 2 turns and moves beside piece 1: the area stays the same.
A square with a smaller square removed can be cut and rearranged into a rectangle with sides (a + b) and (a − b).
a² − b² = (a + b)(a − b)
two squares with a MINUS between them. No middle term
  • EXT Spot it: two terms, both perfect squares, joined by a minus. x² − 49 = x² − 7².
  • EXT Always take out a common factor first. 2x² − 50 = 2(x² − 25).
Worked example

EXT Factorise 25x² − 9.

  1. 25x² = (5x)² and 9 = 3²
  2. a = 5x, b = 3
  3. answer: (5x − 3)(5x + 3)
Worked example

EXT Factorise 2x² − 50.

  1. common factor 2: 2(x² − 25)
  2. x² − 25 = x² − 5²
  3. answer: 2(x − 5)(x + 5)
Worked example

EXT Use algebra to work out 99² − 1 without a calculator.

  1. 99² − 1² = (99 − 1)(99 + 1)
  2. = 98 × 100
  3. = 9800

Factorising by groupingEXTENDED

  • EXT Four terms: group them in pairs, factorise each pair, and look for a common bracket.
Worked example

EXT Factorise xy + 3x + 2y + 6.

  1. group: (xy + 3x) + (2y + 6)
  2. factorise each pair: x(y + 3) + 2(y + 3)
  3. common bracket (y + 3): (x + 2)(y + 3)

4Algebraic fractions EXT

Simplifying

Cancel common FACTORS, never termsx² − 9x² + 3xfactorise(x − 3)(x + 3)x(x + 3)cancelx − 3xWhy? A factor multiplies the whole top. A term is added.x + 33the 3 is a term,so it cannot cancel:x + 3 over 3 ≠ x + 13x + 63factorise first:3(x + 2) ÷ 3= x + 2
Factorise the top and the bottom, then cancel the factors they have in common.
  • EXT You may only cancel factors (things that multiply the whole top or bottom). Never cancel terms (things that are added or subtracted).
  • EXT Multiply: multiply the tops and the bottoms, then cancel. Divide: keep, change, flip, then multiply.
Worked example

EXT Simplify x² − 9x² + 3x.

  1. factorise the top: x² − 9 = (x − 3)(x + 3)
  2. factorise the bottom: x² + 3x = x(x + 3)
  3. cancel the common factor (x + 3)
  4. answer: x − 3x
Worked example

EXT Simplify x3 × 6x².

  1. multiply tops and bottoms: 6x3x²
  2. cancel 3 and one x
  3. answer: 2x
Worked example

EXT Simplify x + 12 ÷ x + 16.

  1. flip and multiply: x + 12 × 6x + 1
  2. cancel (x + 1) and divide 6 by 2
  3. answer: 3

Adding and subtracting

2xmultiply top and bottomby (x + 1)2(x + 1)x(x + 1)3x + 1multiply top and bottomby x3xx(x + 1)Same denominator now: add the tops, keep the bottom2(x + 1) + 3xx(x + 1)=5x + 2x(x + 1)
Make the denominators the same by multiplying top and bottom, then add the numerators.
  • EXT Adding / subtracting: find a common denominator (the product if there is nothing in common), rewrite each fraction, then combine the numerators. Brackets around any subtracted numerator!
Worked example

EXT Write 2x + 3x + 1 as a single fraction.

  1. common denominator: x(x + 1)
  2. 2(x + 1)x(x + 1) + 3xx(x + 1)
  3. add the tops: 2x + 2 + 3x
  4. answer: 5x + 2x(x + 1)
Worked example

EXT Write 3x − 1 − 2x + 2 as a single fraction.

  1. common denominator: (x − 1)(x + 2)
  2. top: 3(x + 2) − 2(x − 1)
  3. = 3x + 6 − 2x + 2
  4. answer: x + 8(x − 1)(x + 2)
Common mistakeWriting 3(x + 2) − 2(x − 1) as 3x + 6 − 2x − 1. The minus multiplies both terms in the bracket: −2(x − 1) = −2x + 2.

5Rearranging formulae

Changing the subject

v = u + at: build v from aa× t+ uvUndo each step in reverse order:v− u÷ taa = (v − u) ÷ t
Build the formula as a chain of operations, then undo them in the reverse order with the opposite operation.

Subject — the letter on its own on one side of a formula. In v = u + at the subject is v.

  • Opposites: + ↔ −, × ↔ ÷, square ↔ square root. Do the same thing to both sides.
  • Method: (1) get the term with the new subject on its own (use + and −); (2) undo any multiplying or dividing; (3) undo powers last.
Worked example

Make t the subject of v = u + at.

  1. subtract u from both sides: v − u = at
  2. divide both sides by a: t = v − ua
Worked example

Make F the subject of C = 59(F − 32).

  1. multiply both sides by 95: 9C5 = F − 32
  2. add 32 to both sides: F = 9C5 + 32
Worked example

Make r the subject of A = πr².

  1. divide by π: Aπ = r²
  2. square root both sides: r = √(Aπ)

When the subject appears twiceEXTENDED

Make x the subject of y = (x + 1) ÷ (x − 2)1y(x − 2) = x + 1multiply both sides by (x − 2)2xy − 2y = x + 1expand the bracket3xy − x = 2y + 1x-terms one side, the rest the other4x(y − 1) = 2y + 1factorise: x now appears just once5x = (2y + 1) ÷ (y − 1)divide by (y − 1)
Get every x-term on one side, factorise so x appears once, then divide.
  • EXT Subject in a fraction: multiply out first. Subject twice: collect all terms with the subject on one side, factorise, then divide.
Worked example

EXT Make x the subject of ax + b = cx + d.

  1. collect x terms on the left: ax − cx = d − b
  2. factorise: x(a − c) = d − b
  3. divide: x = d − ba − c
Worked example

EXT Make x the subject of x + 3y = 2xy.

  1. bring x terms together: x − 2xy = −3y
  2. factorise: x(1 − 2y) = −3y
  3. divide: x = −3y1 − 2y
  4. multiply top and bottom by −1: x = 3y2y − 1
Worked example

EXT Make x the subject of y = √(x + 3).

  1. square both sides: y² = x + 3
  2. subtract 3: x = y² − 3
Exam tipCheck by choosing numbers. For y = (x + 1) ÷ (x − 2): x = 5 gives y = 2. Put y = 2 in the answer (2y + 1) ÷ (y − 1) = 5 ÷ 1 = 5 ✓

6Substitution and proof

Substituting numbers

Evaluate 2x² − 5x + 1 when x = −31replace x with (−3), in brackets2(−3)² − 5(−3) + 12indices first: (−3)² = 92 × 9 − 5(−3) + 13multiply: −5 × −3 = +1518 + 15 + 14add= 34Calculator trapwith the bracket:(−3)² = 9without it:−3² = −9Always bracketa negative.
Replace each letter by its value inside brackets, follow BIDMAS, and take extra care with negatives.
  • Substitute means replace the letters by numbers. Brackets round every negative number stops sign errors.
Worked example

Evaluate s = ut + ½at² when u = 4, a = −2 and t = 3.

  1. ut = 4 × 3 = 12
  2. ½ × (−2) × 3² = ½ × (−2) × 9 = −9
  3. s = 12 + (−9)
  4. = 3
Worked example

The volume of a cone is V = 13πr²h. Find V when r = 3 and h = 5. Give your answer to 3 significant figures.

  1. V = 13 × π × 3² × 5
  2. = 13 × π × 45 = 15π
  3. = 47.12… = 47.1 (3 s.f.)

Algebraic proofEXTENDED

  • EXT To show a statement is always true, use n for any whole number and expand or simplify. Even number: 2n. Odd number: 2n + 1. Consecutive integers: n, n + 1, n + 2.
Worked example

EXT Show that the sum of two consecutive integers is always odd.

  1. let the integers be n and n + 1
  2. sum = n + (n + 1)
  3. = 2n + 1
  4. 2n is even, so 2n + 1 is odd ✓
Worked example

EXT Show that (n + 3)² − (n − 3)² = 12n.

  1. expand: (n² + 6n + 9) − (n² − 6n + 9)
  2. = n² + 6n + 9 − n² + 6n − 9
  3. = 12n ✓

7Test yourself

Test yourself — cover the answers first

Quick check

  1. Simplify 7a + 2b − 3a + b.
  2. Simplify 5x × 3x².
  3. Simplify 20x⁶ ÷ 5x².
  4. Expand 4(3x − 2).
  5. Expand and simplify (x + 4)(x + 5).
  6. Expand (x − 3)².
  7. Factorise 15x − 25.
  8. Factorise 6a²b + 9ab².
  9. Make t the subject of v = u + at.
  10. Make r the subject of A = πr².
  11. Evaluate 3x² − x when x = −2.
  12. Evaluate (a + b)² when a = 3 and b = −5.
  13. EXT Factorise x² + 7x + 12.
  14. EXT Factorise x² − 36.
  15. EXT Factorise 3x² + 10x + 8.
  16. EXT Simplify x² − 25x + 5.
  17. EXT Write x2 + x3 as one fraction.
  18. EXT Expand (2x − 1)(x + 4).

Exam-style

  1. Expand and simplify (3x − 2)(x + 5). [3]
  2. Make m the subject of E = ½mv². [2]
  3. The formula v² = u² + 2as is used with v = 15, u = 5 and s = 10. Find the value of a. [3]
  4. Simplify (2x³y)² ÷ 4xy. [2]
  5. EXT Factorise completely 2x² − 18. [2]
  6. EXT Factorise x² − 5x − 14. [2]
  7. EXT Simplify 2x² + 6xx² − 9. [3]
  8. EXT Write 3x − 2 + 2x + 1 as a single fraction. [3]
  9. EXT Make x the subject of y = x + 2x − 1. [4]
Show answers
  1. 4a + 3b
  2. 15x³
  3. 4x⁴
  4. 12x − 8
  5. x² + 9x + 20
  6. x² − 6x + 9
  7. 5(3x − 5)
  8. 3ab(2a + 3b)
  9. t = (v − u) ÷ a
  10. r = √(A ÷ π)
  11. 14
  12. 4
  13. (x + 3)(x + 4)
  14. (x − 6)(x + 6)
  15. (3x + 4)(x + 2)
  16. x − 5
  17. 5x6
  18. 2x² + 7x − 4
  19. 3x² + 15x − 2x − 10
    = 3x² + 13x − 10
  20. multiply both sides by 2: 2E = mv²
    m = 2E ÷ v²
  21. 15² = 5² + 2 × a × 10
    225 = 25 + 20a
    20a = 200, so a = 10
  22. (2x³y)² = 4x⁶y²
    4x⁶y² ÷ 4xy = x⁵y
  23. take out the 2: 2(x² − 9)
    difference of two squares: 2(x − 3)(x + 3)
  24. numbers that multiply to −14 and add to −5: −7 and +2
    (x − 7)(x + 2)
  25. top: 2x(x + 3)
    bottom: (x − 3)(x + 3)
    cancel (x + 3): 2xx − 3
  26. common denominator (x − 2)(x + 1)
    top: 3(x + 1) + 2(x − 2) = 3x + 3 + 2x − 4
    5x − 1(x − 2)(x + 1)
  27. y(x − 1) = x + 2
    xy − y = x + 2
    xy − x = y + 2
    x(y − 1) = y + 2, so x = y + 2y − 1
Practise this chapter
Algebraic manipulationAlgebraic fractionsIndices II
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