Year 9: Factorising Quadratics Dr J Frost ([email protected]) www.drfrostmaths.com Last modified: 30 th September 2015
Feb 19, 2016
Year 9: Factorising Quadratics
Dr J Frost ([email protected])www.drfrostmaths.com
Last modified: 30th September 2015
Factorising means : To turn an expression into a product of factors.
2 π₯2+4 π₯π§ 2 π₯(π₯+2 π§)
π₯2+3 π₯+2 (π₯+1)(π₯+2)
2x3 + 3x2 β 11x β 6 (2 π₯+1)(π₯β2)(π₯+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 β 2xz x2y β xy2 10xyz β 15x2y xyz β 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 π₯2+4 π₯=2 π₯ (π₯+2 ) π₯2+4 π₯β5=(π₯+5 ) (π₯β1 )
3. Difference of two squares
4 π₯2β1=(2π₯+1 ) (2π₯β1 )
4.
Strategy: either split the middle term, or βgo commandoβ.
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5. Pairwise
π₯3+2π₯2β π₯β2=π₯2 (π₯+2 )β1 (π₯+2 )6. Intelligent Guesswork
? π₯2+ π¦2+2 π₯π¦+π₯+π¦?
TYPE 2:
Expand:
How does this suggest we can factorise say ?
π₯2βπ₯β30=(π₯+5 ) (π₯β6 )
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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:
π₯2β12 π₯+35= (π₯β7 ) (π₯β5 )
π₯2+5 π₯β14=(π₯+7)(π₯β2)
π₯2+6 π₯+5=(π₯+5)(π₯+1)
π₯2+6 π₯+9= (π₯+3 )2
π₯2β6 π₯+9=(π₯β3 )2
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Exercise 21
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π 45
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910
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Six different types of factorisation
1. Factoring out a single term 2.
2 π₯2+4 π₯=2 π₯ (π₯+2 ) π₯2+4 π₯β5=(π₯+5 ) (π₯β1 )
3. Difference of two squares
4 π₯2β1=(2π₯+1 ) (2π₯β1 )
4.
Strategy: either split the middle term, or βgo commandoβ.
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5. Pairwise
π₯3+2π₯2β π₯β2=π₯2 (π₯+2 )β1 (π₯+2 )6. Intelligent Guesswork
? π₯2+ π¦2+2 π₯π¦+π₯+π¦?
TYPE 3: Difference of two squares
Firstly, what is the square root of:
β 4 π₯2=2π₯ β25 π¦ 2=5 π¦
β16 π₯2π¦ 2=4 π₯π¦ βπ₯4 π¦4=π₯2 π¦2
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β9 (π§β6 )2=3(π§β6)?
TYPE 3: Difference of two squares
4 π₯2β9
ΒΏΒΏ
2 π₯2 π₯ 33β β
Click to Start Bromanimation
Quickfire Examples
1βπ₯2=(1+π₯ )(1βπ₯)
π¦ 2β16=(π¦+4 )(π¦β4)
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π₯2π¦ 2β9π2=(π₯π¦+3π ) (π₯π¦β3 π)?
1βπ₯4= (1+π₯2) (1+π₯ ) (1βπ₯ )?
4 π₯2β9 π¦2=(2 π₯+3 π¦ )(2π₯β3 π¦ )?
π₯2β3= (π₯+β3 ) (π₯ββ3 )(Strictly speaking, this is not a valid factorisation)?
Test Your Understanding (Working in Pairs)
(π₯+1 )2β (π₯β1 )2=4 π₯?
49β (1βπ₯ )2=(8β π₯)(6+π₯)?
512β492=200?
18 π₯2β50 π¦2=2 (3 π₯+5 π¦ ) (3 π₯β5 π¦ )?
(2 π‘+1 )2β9 (π‘β6 )2=(5 π‘β17 ) (βπ‘+19 )?
π₯3βπ₯=π₯ (π₯+1)(π₯β1)?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. ππ+ππ= ππ which is also a prime factor. ππ+ππ= + = ππ ππ ππ which is prime. ππ+ππ=ππππ. 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 π₯2+π₯β3Factorise using:
a. βGoing commandoβ* b. Splitting the middle term
* Not official mathematical terminology.
Essentially βintelligent guessingβ of the two brackets, by considering what your guess would expand to.
(2 π₯+3)(π₯β1)? ?? ?
How would we get the term in the expansion?
How could we get the -3?
2 π₯2+π₯β3β1Unlike before, we want two numbers which multiply to give the first times the last number.
2 π₯2+3 π₯β2π₯β3Factorise first and second half separately.
βSplit the middle termβ
ΒΏ π₯ (2 π₯+3 )β1(2 π₯+3)ΒΏ (2π₯+3)(π₯β1)Thereβs a
common factor of
More Examples
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Exercise 41
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1011
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βCommandoβ 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.
π₯2β4 π₯=π (πβπ ) π₯2+7 π₯β30=(π+ππ ) (πβπ )
3. Difference of two squares
9β16 π¦2= (π+π π ) (πβπ π )
4.
Strategy: either split the middle term, or βgo commandoβ.
5. Pairwise
π₯3+2π₯2β π₯β2=π₯2 (π₯+2 )β1 (π₯+2 )6. Intelligent Guesswork
π₯2+ π¦2+2 π₯π¦+π₯+π¦
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RECAP ::
Method A: Guessing the brackets Method B: Splitting the middle term
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This method of βintelligent guessingβ 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 βpairwise factorisationβ.
Both of these methods can be extended to more general expressions.
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TYPE 5: Intelligent Guessing
π₯2+ππ₯+ππ₯+ππ
ΒΏ ()()
Just think what brackets would expand to give you expression. Look at each term one by one.
π₯ π₯+π +πIt works!
ππβπ+πβ1?
This factorisation will become particularly important when we cover something called βDiophantine Equationsβ.
Test Your Understanding
π₯π¦+3 π₯β2 π¦β6=(πβπ ) (π+π )Bro Tip: The arose because of collecting like terms in the expansion. It might therefore be easier to first think how we get the βeasierβ 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 thereβs an βalgebraic symmetryβ in and , as and could be swapped without changing the expression. But thereβs 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.
π₯2β π¦2+4 π₯+4 π¦π₯3β2π₯2βπ₯+2
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π₯2+ππ₯+ππ₯+ππ=ΒΏ??
Test Your Understanding
π₯2βπ₯π¦+2 π₯β2 π¦?
ππ+π+π+1?
π₯3β3π₯2β4 π₯+12π2+π2+2ππ+ππ+ππCan you split the terms
into two blocks, where in each block you can factorise?
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Challenge Wall!
π₯π¦β π₯β π¦+1=(πβπ)(πβπ)
π₯2+ π¦2+2 π₯π¦β1=(π+π+π)(π+π βπ)
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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.
π₯ π¦ 2+3 π¦2+π₯+3=(π+π)(ππ+π)
π₯3+2π₯2β9 π₯β18=(π+π)(πβπ)(π+π)
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Warning: Pairwise factorisation doesnβt always work. You sometimes have to resort to βintelligent guessingβ.
Exercise 5Factorise the following using either βpairwise factorisationβ or βintelligent guessingβ.
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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
Itβs fine to factorise out an entire expression:
π₯ (π₯+2 )β3 (π₯+2 )β(π₯+2)(π₯β3)
π₯ (π₯+1 )2+2 (π₯+1 )β (π₯2+π₯+2 ) (π₯+1 )
2 (2π₯β3 )2+π₯ (2π₯β3 )β(5 π₯β6 )(2π₯β3)π (2π+1 )+π (2π+1 )β(π+π)(2π+1)
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