Algebra Factorising Linear and Quadratics Skipton Girls High

Algebra: Factorising- Linear and Quadratics Skipton Girls’ High School

Factors What does the factor of a number mean? ? Numbers which divide the original number without a remainder. Factors of 8: 1, 2, 4, 8 ? Factors of 2 x: 1, 2, x, 2 x? Factors of 2 x 2: ? 2, 2 x 2 1, 2, x, 2 x, x

Factorising is the reverse of expanding. When you have a sum of terms, just identify the common factor. i. e. Find the largest expression each of your terms is divisible by. Common factor = 2 ? 2 x + 4 ? So 2 x + 4 = 2(x + 2) (You could always check this by expanding out the brackets)

Factorising is the reverse of expanding. When you have a sum of terms, just identify the common factor. i. e. Find the largest expression each of your terms is divisible by. Common factor = 3 x? 3 x 2 + 9 x ? So 3 x 2 + 9 x = 3 x(x + 3) We could have just ‘factored out’ the 3, but we wouldn’t have fully factorised because there’s also a factor of x.

Factorising xy + x ? = x(y + 1) ? 2 xy + 4 x = 2 x(y + 2)

Exercise 1 ? ? ? ?

Exercise 2 ? ? ? 16 factors: 1, 2, x, y, z, 2 x, 2 y, 2 z, xy, xz, yz, 2 xy, 2 xz, 2 yz, xyz, 2 xyz 36 factors ? ? 2 can either appear in the factor or not (2 possibilities) x can either appear 0 times, 1 time, up to a times (a + 1 possibilities) y similarly has b + 1 possibilities and z has c +1 possibilities. So 2(a + 1)(b + 1)(c + 1) possible factors. ?

Dealing with fractions When factorising, it’s convention to have any fractions outside the bracket. ? ? Bro Tip: Make sure the fractions have a common denominator.

Test Your Understanding ? ? ?

Exercise 3 ? ? ? ? ?

Factorising Overview Factorising means : To turn an expression into a product of factors. Year 8 Factorisation So what factors can we see here? Factorise Year 9 Factorisation A Level Factorisation 2 x 3 + 3 x 2 – 11 x – 6 Factorise

Six different types of factorisation 1. Factoring out a single term ? 3. Difference of two squares ? 5. Pairwise ? ? 6. Intelligent Guesswork ? ?

? ? ? Tip: Think of the factor pairs of 30. You want a pair where the sum or difference of the two numbers is the middle number (-1).

A few more examples: ? ? ?

Exercise 2 1 2 3 ? ? ? 5 6 7 8 9 10 Hardcore N 2 N 3 12 ? ? ? ? ? ? 4 N 1 11 13 14 ? ? N 4 N 5 ? ? ?

Six different types of factorisation 1. Factoring out a single term 3. Difference of two squares ? 5. Pairwise ? 6. Intelligent Guesswork ? ?

TYPE 3: Difference of two squares Firstly, what is the square root of: ? ? ?

TYPE 3: Difference of two squares Click to Start animation

Quickfire Examples ? ? (Strictly speaking, this is not a valid factorisation)

Test Your Understanding (Working in Pairs) Tip: Sometimes you can use one type of factorisation followed by another. Perhaps common term first? ? ? ? ? ?

Exercise 3 1 ? ? 2 N 3 ? 3 4 ? 5 6 ? 7 8 9 N 4 ? 10 ? ? N 1 N 2 ? ?

Factorise using: b. Splitting the middle term a. ‘Going commando’* Essentially ‘intelligent guessing’ of the two brackets, by considering what your guess would expand to. ? ? ‘Split the middle term’ ? ? Unlike before, we want two numbers which multiply to give the first times the last number. Factorise first and second half separately. How could we get the -3? * Not official mathematical terminology.

More Examples ? ?

Exercise 4 1 2 3 4 5 6 7 8 ? ? ? ? 9 10 11 N 1 N 2 ? ? ?

RECAP : : Six different types of factorisation 1. Factoring out a single term ? 3. Difference of two squares ? 5. Pairwise 6. Intelligent Guesswork

Method A: Guessing the brackets Method B: Splitting the middle term ? ? Both of these methods can be extended to more general expressions. This method of ‘intelligent guessing’ can be extended to non-quadratics. After we split the middle term, we looked at the expression in two pairs and factorised. I call more general usage of this ‘pairwise factorisation’.

TYPE 5: Intelligent Guessing Just think what brackets would expand to give you expression. Look at each term one by one. It works! This factorisation will become particularly important when we cover something called ‘Diophantine Equations’. ?

Test Your Understanding 1 ? ? ? 2 3 N ?

TYPE 6: Pairwise Factorisation We saw earlier with splitting the middle term that we can factorise different parts of the expression separately and hope that a common term emerges. ? ? ? ?

Test Your Understanding 1 ? 2 ? 3 ? Can you split the terms into two blocks, where in each block you can factorise? N ?

Challenge Wall! 1 2 3 4 Instructions: Divide your paper into four. Try and get as far up the wall as possible, then hold up your answers for me to check. Use any method of factorisation. Warning: Pairwise factorisation doesn’t always work. You sometimes have to resort to ‘intelligent guessing’. 4 ? 3 ? 2 ? 1 ?

Exercise 5 Factorise the following using either ‘pairwise factorisation’ or ‘intelligent guessing’. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ? ? ? ? ? ? N 1 N 2 ? ? ? N 3 N 4 N 5 ? ? ?

Summary For the following expressions, identify which of the following factorisation techniques that we use, out of: (it may be multiple!) 1 2 3 4 5 6 ? ? ? ? ?

Factorising out an expression It’s fine to factorise out an entire expression: ? ?