Algebraic manipulation
Section: Algebra and graphs 1 | Syllabus: Cambridge IGCSE Mathematics (0580)
Collecting Like Terms and Expanding Brackets
Simplifying an expression means writing it with as few terms as possible. Collecting like terms combines terms of the same type, and expanding brackets removes them by multiplying out.
- Like terms have exactly the same letters raised to the same powers, e.g. 3x and 7x, or 2y² and -5y²
- To expand a single bracket, multiply every term inside it by the term outside
- To expand two brackets, multiply every term in the first bracket by every term in the second, then collect like terms
Worked Example: Collecting Like Terms
- Question: Simplify.
5x + 3y - 2x + 7y- Step 1: Group the x terms and the y terms: (5x - 2x) + (3y + 7y)
- Answer: 3x + 10y
Worked Example: Expanding Double Brackets
- Question: Expand the brackets and simplify.
(3x + 2)(x + 5)- Step 1: Multiply every term in the first bracket by every term in the second: 3x · x + 3x · 5 + 2 · x + 2 · 5
- Step 2: = 3x^2 + 15x + 2x + 10
- Step 3: Collect the like terms: 3x^2 + 17x + 10
- Answer: 3x² + 17x + 10
Common Mistakes
MistakeCombining unlike terms, e.g. simplifying 5x + 3y as 8xy or 8(x+y)
Fixonly terms with exactly the same letters and powers can combine - 5x and 3y are different types of term and must stay separate
MistakeMissing one of the four cross-products when expanding double brackets, e.g. forgetting the "inner" or "outer" term
Fixevery term in the first bracket must multiply every term in the second bracket - that is always 4 products for two brackets of 2 terms each
Factorising by Common Factors
Factorising is the reverse of expanding: writing an expression as a product of brackets or terms instead of a sum. The simplest kind pulls out a factor common to every term.
- Find the highest common factor (HCF) of every term, including any common letters
- Write the HCF outside a bracket, with each original term divided by the HCF inside
- Check by expanding the answer back out - it should return the original expression
Worked Example: Factorising by a Common Factor
- Question: Factorise.
12x³ - 8x- Step 1: The HCF of 12x³ and 8x is 4x
- Step 2: Divide each term by 4x: 12x^3 ÷ 4x = 3x^2, 8x ÷ 4x = 2
- Answer: 4x(3x² - 2)
Common Mistakes
MistakeNot taking out the highest common factor, e.g. factorising 12x³ - 8x as 2x(6x² - 4)
Fix2x(6x² - 4) still has a common factor of 2 left inside the bracket - always check the HCF has been fully extracted: 4x(3x² - 2)
MistakeLosing a sign when factorising a subtraction, e.g. writing 4x(3x² + 2) instead of 4x(3x² - 2)
Fixexpand the answer back out to check it matches the original expression exactly, including signs
Expanding More Complex Products
Extended Only
When more than two brackets are multiplied together, expand two of them first, then multiply the result by what remains, collecting like terms at the very end.
- Expand a repeated bracket like (x - 4)² by writing it out in full as (x - 4)(x - 4) first - it is not the same as x² + 16
- Multiply the result by any remaining bracket, then collect like terms once at the end
The area of the whole square (a+b)² equals the sum of its four regions: a² + ab + ab + b² = a² + 2ab + b²
Worked Example: Expanding Three Brackets
- Question: Expand and simplify (x + 2)(x - 4)².
- Step 1: Expand the squared bracket first: (x-4)^2 = x^2 - 8x + 16
- Step 2: Multiply by the remaining bracket: (x+2)(x^2-8x+16) = x^3 - 8x^2 + 16x + 2x^2 - 16x + 32
Every term in (x+2) multiplies every term in the trinomial - six products in total before collecting like terms
- Question (continued):
- Step 3: Collect like terms: x^3 - 6x^2 + 0x + 32
- Answer: x³ - 6x² + 32
Common Mistakes
MistakeExpanding (x - 4)² as x² + 16, missing the middle term entirely
Fixa squared bracket must be written out and expanded like any double bracket: (x-4)² = (x-4)(x-4) = x² - 8x + 16
MistakeLosing a term when collecting like terms across a longer expansion
Fixlist every term from the expansion first, then group by matching powers of x one power at a time (x³, then x², then x, then the constant)
Factorising Quadratics and Special Forms
Extended Only
Quadratics and four-term expressions can usually be factorised into two brackets. Two special patterns - a difference of two squares, and a perfect square - can be spotted and factorised instantly, without splitting any terms.
- To factorise ax^2+bx+c, find two numbers that multiply to give a × c and add to give b, then split the middle term and factorise by grouping
- A four-term expression like ax+bx+kay+kby can be factorised by grouping: pair the terms so each pair shares a common factor, factorise each pair, then factor out the common bracket
- Difference of two squares: a^2 - b^2 = (a-b)(a+b)
- Perfect square: a^2 + 2ab + b^2 = (a+b)^2
Worked Example: Factorising ax² + bx + c
- Question: Factorise.
3x² + 5x - 12- Step 1: Find two numbers with product 3 × (-12) = -36 and sum 5: these are 9 and -4
- Step 2: Split the middle term: 3x^2 + 9x - 4x - 12
- Step 3: Factorise by grouping: 3x(x+3) - 4(x+3) = (3x-4)(x+3)
- Answer: (3x - 4)(x + 3)
Worked Example: Factorising by Grouping
- Question: Factorise.
px + 5p - 2qx - 10q- Step 1: Group into pairs that share a factor: (px+5p) - (2qx+10q)
- Step 2: Factorise each pair: p(x+5) - 2q(x+5)
- Step 3: Factor out the common bracket: (p-2q)(x+5)
- Answer: (p - 2q)(x + 5)
Worked Example: Difference of Two Squares and Perfect Squares
- Question: Factorise.
(i) 9x² - 25 (ii) x² + 10x + 25- (i): Recognise as a difference of two squares: 9x^2 - 25 = (3x)^2 - 5^2 = (3x-5)(3x+5)
- (ii): Recognise as a perfect square, since 5^2=25 and 2 × 5 = 10: x^2+10x+25 = (x+5)^2
- Answer: (i) (3x-5)(3x+5), (ii) (x+5)²
Common Mistakes
MistakeSwapping the two conditions when splitting the middle term, e.g. looking for numbers that add to give a × c and multiply to give b
Fixthe two numbers must multiply to a × c and add to b - check both conditions before splitting the middle term
MistakeEnding up with two different brackets after grouping, e.g. p(x+5) - 2q(x-5), and not noticing the grouping has gone wrong
Fixboth brackets after grouping must be identical - if they are not, re-check the signs used when factorising each pair
MistakeTrying to factorise a sum of two squares, e.g. attempting to factorise x² + 25
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