Year 9: Factorising Quadratics

Dr J Frost (jfrost@tiffin.kingston.sch.uk)www.drfrostmaths.com

Last modified: 30th September 2015

Factorising means : To turn an expression into a product of factors.

2 𝑥2+4 𝑥𝑧 2 𝑥(𝑥+2 𝑧)

𝑥2+3 𝑥+2 (𝑥+1)(𝑥+2)

2x3 + 3x2 – 11x – 6 (2 𝑥+1)(𝑥−2)(𝑥+3)

Year 8 Factorisation

Year 9 Factorisation

A Level Factorisation

Factorise

Factorise

Factorise

So what factors can we see here?

Factorising Overview

5 + 10x x – 2xz x2y – xy2 10xyz – 15x2y xyz – 2x2yz2 + x2y2

Starter

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Exercise 1

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Note: We tend to factorise any fraction out, e.g.

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Six different types of factorisation

1. Factoring out a single term 2.

2 𝑥2+4 𝑥=2 𝑥 (𝑥+2 ) 𝑥2+4 𝑥−5=(𝑥+5 ) (𝑥−1 )

3. Difference of two squares

4 𝑥2−1=(2𝑥+1 ) (2𝑥−1 )

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Strategy: either split the middle term, or ‘go commando’.

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5. Pairwise

𝑥3+2𝑥2− 𝑥−2=𝑥2 (𝑥+2 )−1 (𝑥+2 )6. Intelligent Guesswork

? 𝑥2+ 𝑦2+2 𝑥𝑦+𝑥+𝑦?

TYPE 2:

Expand:

How does this suggest we can factorise say ?

𝑥2−𝑥−30=(𝑥+5 ) (𝑥−6 )

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Bro 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).

and add to give 3.

and times to give 2.

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TYPE 2:

A few more examples:

𝑥2−12 𝑥+35= (𝑥−7 ) (𝑥−5 )

𝑥2+5 𝑥−14=(𝑥+7)(𝑥−2)

𝑥2+6 𝑥+5=(𝑥+5)(𝑥+1)

𝑥2+6 𝑥+9= (𝑥+3 )2

𝑥2−6 𝑥+9=(𝑥−3 )2

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Exercise 21

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Six different types of factorisation

1. Factoring out a single term 2.

2 𝑥2+4 𝑥=2 𝑥 (𝑥+2 ) 𝑥2+4 𝑥−5=(𝑥+5 ) (𝑥−1 )

3. Difference of two squares

4 𝑥2−1=(2𝑥+1 ) (2𝑥−1 )

4.

Strategy: either split the middle term, or ‘go commando’.

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5. Pairwise

𝑥3+2𝑥2− 𝑥−2=𝑥2 (𝑥+2 )−1 (𝑥+2 )6. Intelligent Guesswork

? 𝑥2+ 𝑦2+2 𝑥𝑦+𝑥+𝑦?

TYPE 3: Difference of two squares

Firstly, what is the square root of:

√ 4 𝑥2=2𝑥 √25 𝑦 2=5 𝑦

√16 𝑥2𝑦 2=4 𝑥𝑦 √𝑥4 𝑦4=𝑥2 𝑦2

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√9 (𝑧−6 )2=3(𝑧−6)?

TYPE 3: Difference of two squares

4 𝑥2−9

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2 𝑥2 𝑥 33√ √

Click to Start Bromanimation

Quickfire Examples

1−𝑥2=(1+𝑥 )(1−𝑥)

𝑦 2−16=(𝑦+4 )(𝑦−4)

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𝑥2𝑦 2−9𝑎2=(𝑥𝑦+3𝑎 ) (𝑥𝑦−3 𝑎)?

1−𝑥4= (1+𝑥2) (1+𝑥 ) (1−𝑥 )?

4 𝑥2−9 𝑦2=(2 𝑥+3 𝑦 )(2𝑥−3 𝑦 )?

𝑥2−3= (𝑥+√3 ) (𝑥−√3 )(Strictly speaking, this is not a valid factorisation)?

Test Your Understanding (Working in Pairs)

(𝑥+1 )2− (𝑥−1 )2=4 𝑥?

49− (1−𝑥 )2=(8− 𝑥)(6+𝑥)?

512−492=200?

18 𝑥2−50 𝑦2=2 (3 𝑥+5 𝑦 ) (3 𝑥−5 𝑦 )?

(2 𝑡+1 )2−9 (𝑡−6 )2=(5 𝑡−17 ) (−𝑡+19 )?

𝑥3−𝑥=𝑥 (𝑥+1)(𝑥−1)?Bro Tip: Sometimes you can use one type of factorisation followed by another. Perhaps common term first?

Exercise 3

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Find four prime numbers less than 100 which are factors of (Hint: you can keep factorising!)

So clearly 5 is a factor. 𝟑𝟐+𝟐𝟐= 𝟏𝟑 which is also a prime factor. 𝟑𝟒+𝟐𝟒= + = 𝟖𝟏 𝟏𝟔 𝟗𝟕 which is prime. 𝟑𝟖+𝟐𝟖=𝟔𝟖𝟏𝟕. This fails all the divisibility tests for the primes up to 11, and dividing by 13 (by normal division) fails, but dividing by 17 (again by normal division) works, giving us our fourth prime. (Alternatively, noting that , then , so 17 is a factor)

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N3 [IMO] What is the highest power of 2 that is a factor of ?

So the highest power is 8.?

TYPE 4:

2 𝑥2+𝑥−3Factorise using:

a. ‘Going commando’* b. Splitting the middle term

* Not official mathematical terminology.

Essentially ‘intelligent guessing’ of the two brackets, by considering what your guess would expand to.

(2 𝑥+3)(𝑥−1)? ?? ?

How would we get the term in the expansion?

How could we get the -3?

2 𝑥2+𝑥−3⊕1Unlike before, we want two numbers which multiply to give the first times the last number.

2 𝑥2+3 𝑥−2𝑥−3Factorise first and second half separately.

‘Split the middle term’

¿ 𝑥 (2 𝑥+3 )−1(2 𝑥+3)¿ (2𝑥+3)(𝑥−1)There’s a

common factor of

More Examples

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Exercise 41

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‘Commando’ starts to become difficult from this question onwards because the coefficient of is not prime.

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RECAP :: Six different types of factorisation

1. Factoring out a single term 2.

𝑥2−4 𝑥=𝒙 (𝒙−𝟒 ) 𝑥2+7 𝑥−30=(𝒙+𝟏𝟎 ) (𝒙−𝟑 )

3. Difference of two squares

9−16 𝑦2= (𝟑+𝟒 𝒚 ) (𝟑−𝟒 𝒚 )

4.

Strategy: either split the middle term, or ‘go commando’.

5. Pairwise

𝑥3+2𝑥2− 𝑥−2=𝑥2 (𝑥+2 )−1 (𝑥+2 )6. Intelligent Guesswork

𝑥2+ 𝑦2+2 𝑥𝑦+𝑥+𝑦

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Method A: Guessing the brackets Method B: Splitting the middle term

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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’.

Both of these methods can be extended to more general expressions.

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TYPE 5: Intelligent Guessing

𝑥2+𝑎𝑥+𝑏𝑥+𝑎𝑏

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Just think what brackets would expand to give you expression. Look at each term one by one.

𝑥 𝑥+𝑎 +𝑏It works!

𝑎𝑏−𝑎+𝑏−1?

This factorisation will become particularly important when we cover something called ‘Diophantine Equations’.

Test Your Understanding

𝑥𝑦+3 𝑥−2 𝑦−6=(𝒙−𝟐 ) (𝒚+𝟑 )Bro Tip: The arose because of collecting like terms in the expansion. It might therefore be easier to first think how we get the ‘easier’ terms like the (where the coefficient of the term is 1) when we try to fill in the brackets.

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Bro Tip: Notice that there’s an ‘algebraic symmetry’ in and , as and could be swapped without changing the expression. But there’s an asymmetry in .This gives hints about the factorisation, as the same symmetry must be seen.

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

𝑥2− 𝑦2+4 𝑥+4 𝑦𝑥3−2𝑥2−𝑥+2

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𝑥2+𝑎𝑥+𝑏𝑥+𝑎𝑏=¿??

Test Your Understanding

𝑥2−𝑥𝑦+2 𝑥−2 𝑦?

𝑎𝑏+𝑎+𝑏+1?

𝑥3−3𝑥2−4 𝑥+12𝑎2+𝑏2+2𝑎𝑏+𝑎𝑐+𝑏𝑐Can you split the terms

into two blocks, where in each block you can factorise?

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Challenge Wall!

𝑥𝑦− 𝑥− 𝑦+1=(𝒙−𝟏)(𝒚−𝟏)

𝑥2+ 𝑦2+2 𝑥𝑦−1=(𝒙+𝒚+𝟏)(𝒙+𝒚 −𝟏)

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3 4Instructions: 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.

𝑥 𝑦 2+3 𝑦2+𝑥+3=(𝒙+𝟑)(𝒚𝟐+𝟏)

𝑥3+2𝑥2−9 𝑥−18=(𝒙+𝟑)(𝒙−𝟑)(𝒙+𝟐)

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Warning: Pairwise factorisation doesn’t always work. You sometimes have to resort to ‘intelligent guessing’.

Exercise 5Factorise the following using either ‘pairwise factorisation’ or ‘intelligent guessing’.

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SummaryFor the following expressions, identify which of the following factorisation techniques that we use, out of: (it may be multiple!)

Factorising out single term: 1

Simple quadratic factorisation: 2

Difference Of Two Squares: 3

Commando/Splitting Middle Term: 4

(1)(3)(1), (3)

(2)(4)(2), (6)(5)(1), (2)

(5) or (6) (1), (3)

Pairwise: 5Intelligent Guesswork: 6

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Factorising out an expression

It’s fine to factorise out an entire expression:

𝑥 (𝑥+2 )−3 (𝑥+2 )→(𝑥+2)(𝑥−3)

𝑥 (𝑥+1 )2+2 (𝑥+1 )→ (𝑥2+𝑥+2 ) (𝑥+1 )

2 (2𝑥−3 )2+𝑥 (2𝑥−3 )→(5 𝑥−6 )(2𝑥−3)𝑎 (2𝑐+1 )+𝑏 (2𝑐+1 )→(𝑎+𝑏)(2𝑐+1)

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