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