Expanding and Factorising
Thats Mathematics - You Tube.mp4
Expanding
Can’t cannot
What is ‘expanding’?
Expanding
Yes, that is expanding
But what we want to know is how to expand an algebraic equation
3(7 + 10)
5x(6y – 7z)
4(x – 4) + 5
ExpandingExpansion means to multiply
everything inside the brackets by what is directly outside the brackets
Think WriteWrite the expression
Expand the brackets
Multiply out the brackets
4(x – 4) + 5
= 4(x) + 4(-4) + 5= 4x -16 + 5
Expanding single bracketsAfter expanding brackets, simplify by
collecting any like terms
Think
Collect any like terms
= 4x -16 + 5
= 4x - 11
Write
Your turn.......
Remember:Stop and think
Ask your neighbour quietly
Hand-up
Move on until I can get to you
Some music as you work
Got a question
Expanding two bracketsExpand each bracket: working from
left to right
Expanding pairs of bracketsWhen multiplying expressions within
pairs of brackets, multiply each term in the first bracket by each term in the second bracket, then collect the like terms
Expanding pairs of bracketsYou can use the ‘FOIL’ method to help
you keep track of which terms are to be multiplied together
First – multiply the first term in each bracket Outer – multiply the 2 outer terms Inner – multiply the 2 inner terms Last – multiply the last term of each bracket
Expansion patternsDifference of two squares
(a + b) (a – b) = a2 – b2
Expansion patternsPerfect squares (identical brackets)
Square the first term, add the square of the last term, then add (or subtract) twice their product
(a + b) (a + b) = a2 + 2ab + b2
(a – b) (a – b) = a2 – 2ab – b2
Expanding more than two bracketsBrackets or pairs of brackets that are
added or subtracted must be expanded separately
Always collect any like terms following an expansion
Factorising Factorising is the ‘opposite’ of
expanding, going from an expanded form to a more compact form
Factor pairs of a term are numbers and pronumerals which, when multiplied together, produce the original term
Highest Common FactorThe number itself and 1 are factors of
every integer
The highest common factor (HCF) of given terms is the largest factor that divides into all terms without a remainder
Factorising using the highest common factorAn expression is factorised by finding
the HCF of each term, dividing it into each term and placing the result inside the brackets, with the HCF outside the brackets
Factorising using the difference of two squares ruleTo factorise a difference of two squares,
a2 – b2
we use the rule or formula
(a + b) (a – b) = a2 – b2
in reverse
a2 – b2 = (a + b) (a – b)
Factorising using the Difference-of-two-squares ruleLook for the common factor first
If there is one, factorise by taking it out
Rewrite the expression showing the two squares and identifying the a and b parts of the expression
Factorising using the Difference-of-two-squares ruleFactorise, using the rule
a2 – b2 = (a + b) (a – b)
Simplifying algebraic fractions
Factorise the numerator and the denominator
Cancel factors where appropriate
Simplifying algebraic fractions
If two fractions are multiplied, factorise where possible then cancel any factors, one from the numerator and one from the denominator
Simplifying algebraic fractions
If two fractions are divided, remember to multiply the reciprocal of the second fraction before factorising and cancelling
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