Manipulating Algebraic Expressions



Estimated Time: 1 hour 15 mins

Collecting Terms



Sometimes you are told to "simplify" an expression, or you want to tidy it up so it's easier to read. Collecting 'like' terms is a simple way of doing this - you need to know how to do this before you even start AS Maths.

'Like' terms have the same variable and 'unlike' terms don't.
The power of the variable must be the same too - e.g. 4x and 5x2 are unlike terms so they can't be collected.

Example
Simplify the expression 2x + 4y - 5z + 5x - 9y + 2z + 4x - 7y + 8z.
                                      = 2x + 4x - 5x + 4y - 9y - 7y + 2z + 8z - 5z          Collect like terms.
                                      = 6x - 5x + 4y - 16y + 10z - 5z                                     Tidy up
                                      = x - 12y + 5z


Removing Brackets



Sometimes you need to remove the brackets of an equation before you can collect the like terms.

Examples
Simplify the expression 3(2x - 4y) - 4(x - 5y).
                                      = 6x - 12y - 4x + 20y                            Open the brackets.
                                      = 6x - 4x + 20y - 12y                                     Collect like terms
                                      = 2x + 8y

Simplify the expression x(x + 2) - (x - 4).
                                      = x2 + 2x - x + 4          Open the brackets.
                                      = x2 + x + 4

Simplify the expression a(b + c) - ac.
                                      = ab + ac - ac          Open the brackets.
                                      = ab


Factorisation



If you factorise an expression, it can be easier to use and neater to write or can help you to interpret its meaning. It's the reverse process of multiplying out brackets (putting in brackets). If there is a common factor (the HCF of all terms), it can be taken outside a bracket.

Examples
Factorise 12x - 18y.
                                      = 6(2x - 3y)          6 is a factor of 12 and 18.

Factorise x2 - 2xy + 3xz.
                                      = x(x - 2y + 3z)          x is a factor of all the terms.


Multiplication



Example
Multiply 3p2qr × 4pq3 × 5qr2.
                                      = 3 × 4 × 5 × p2 × p × q × q3 × q × r × r2
                                      = 60 p3 × q5× r3
                                      = 60p3q5r3
Fractions



The rules for algebraic fractions are the same as arithmetic fractions. When adding, they need a common denominator just like arithmetic fractions do. In the example below, we can see this.

Examples
Simplify x⁄2 - 2y⁄10 + z⁄4
                                      = 10x⁄20 - 4y⁄20 + 5z⁄20
                                      = 10x - 4y + 5z
                                                 20




Simplify x2⁄y × y2⁄x
                                      = x3⁄xy - y3⁄xy
                                      = x3 - y3
                                            xy


Simplify 3x2⁄5y - 5yz⁄6x
                                      = xz⁄2


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



Simplify = 24x + 6
              3(4x + 1)
                                      = 6(4x + 1)
                                         3(4x + 1)
= 2



Exercises
Exam Questions

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