gcse: quadratic functions and simplifying rational expressions
DESCRIPTION
GCSE: Quadratic Functions and Simplifying Rational Expressions. Dr J Frost ([email protected]) . Last modified: 25 th August 2013. Factorising Overview. Factorising means : To turn an expression into a product of factors. So what factors can we see here?. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/1.jpg)
GCSE: Quadratic Functions and Simplifying Rational Expressions
Dr J Frost ([email protected])
Last modified: 10th December 2015
![Page 2: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/2.jpg)
Factorising means : To turn an expression into a product of factors.
2x2 + 4xz 2x(x+2z)
x2 + 3x + 2 (x+1)(x+2)
2x3 + 3x2 – 11x – 6 (2x+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
![Page 3: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/3.jpg)
5 + 10x x – 2xz x2y – xy2 10xyz – 15x2y xyz – 2x2yz2 + x2y2
Factor Challenge
![Page 4: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/4.jpg)
1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
??
???
???
??
??
Extension Question:What integer (whole number) solutions are there to the equation
Answer: . So the two expressions we’re multiplying can be This gives solutions of ?
Exercises
![Page 5: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/5.jpg)
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)
?
?
?
?
![Page 6: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/6.jpg)
Harder Factorisation
?
?
![Page 7: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/7.jpg)
Exercises
Edexcel GCSE Mathematics Textbook
Page 111 – Exercise 8DQ1 (right column), Q2 (right column)
![Page 8: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/8.jpg)
Expanding two brackets
1
2
3
4
5
6
78
9
10
1112
13
14
???
???
??
??
??
??
![Page 9: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/9.jpg)
Faster expansion of squared brackets
There’s a quick way to expand squared brackets involving two terms:
?
?
?
?
![Page 10: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/10.jpg)
Four different types of factorisation
1. Factoring out a 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’.
? ?
? ?
![Page 11: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/11.jpg)
2.
Which is ?
How does this suggest we can factorise say ?
𝑥2−𝑥−30=(𝑥+5 ) (𝑥−6 )?
Is there a good strategy for working out which numbers to use?
![Page 12: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/12.jpg)
2.
1
2
3
𝝅45
67
8
910
???
?????
??
?
![Page 13: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/13.jpg)
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
? ?
? ?
√9 (𝑧−6 )2=3(𝑧−6)?
![Page 14: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/14.jpg)
3. Difference of two squares
4 𝑥2−9
¿¿
2 𝑥2 𝑥 33√ √
Click to Start Bromanimation
![Page 15: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/15.jpg)
3. Difference of two squares
1−𝑥2=(1+𝑥 )(1−𝑥)
(𝑥+1 )2− (𝑥−1 )2=4 𝑥
?
?
49− (1−𝑥 )2=(8− 𝑥)(6+𝑥)?
512−492=200?(In your head!)
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 )?
![Page 16: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/16.jpg)
3. Difference of two squares
Exercises:
1
2
3
4
5
6
7
8
9
10
??
???
???
??
![Page 17: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/17.jpg)
4.
2 𝑥2+𝑥−3=(2 𝑥+3)(𝑥−1)Factorise using:
a. The ‘commando’ method*
b. Splitting the middle term
* Not official mathematical terminology.
?
![Page 18: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/18.jpg)
4.
?
?
?
?
![Page 19: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/19.jpg)
Exercises
Well Hardcore:
1
2
3
4
5
6
7
8
9
1011
NN
???????
??
??
‘Commando’ starts to become difficult from this question onwards.
??
![Page 20: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/20.jpg)
Simplifying Algebraic Fractions
2𝑥2+4 𝑥𝑥2−4
=2 𝑥𝑥−2
3 𝑥+3𝑥2+3 𝑥+2
=3
𝑥+2
2𝑥2−5𝑥−36 𝑥3−2𝑥4
=− 2𝑥+12𝑥3
?
?
?
![Page 21: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/21.jpg)
Negating a difference
?
?
?
?
![Page 22: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/22.jpg)
Exercises
1
2
3
4
5
6
7
8
9
10
11
?
?
?
?
?
?
?
?
?
?
?
![Page 23: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/23.jpg)
Algebraic Fractions
35 +
110=
710
23−
14=
512
?
?
(Note: If you’ve added/subtracted fractions before using some ‘cross-multiplication’-esque method, unlearn it now, because it’s pants!)
How did we identify the new denominator to use?
![Page 24: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/24.jpg)
Algebraic Fractions
The same principle can be applied to algebraic fractions.
1𝑥 +
2𝑥2
=𝑥𝑥2
+2𝑥2
=𝑥+2𝑥2?
1𝑥 −
2𝑥2+2𝑥
=1
𝑥+2?
![Page 25: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/25.jpg)
The Wall of Algebraic Fraction Destiny
“To learn the secret ways of algebra ninja, simplify fraction you must.”
13𝑥+6 +
15 𝑥+10−
215 𝑥+30=
25 (𝑥+2 )
52𝑥+1
− 32 𝑥+3
=4 (𝑥+3 )
(2𝑥+1 ) (2𝑥+3 )
1𝑥+1−
1𝑥=− 1
𝑥+1?
?
?
![Page 26: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/26.jpg)
Recap
?
?
?
?
?
![Page 27: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/27.jpg)
Exercises
?
?
?
?
?
?
?
?
?
?
?
1
2
3
4
5
6
7
8
9
10
11
![Page 28: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/28.jpg)
Expand the following:
(𝑥+3 )2=𝑥2+6 𝑥+9
(𝑥+5 )2+1=𝑥2+10 𝑥+26
(𝑥−3 )2=𝑥2−6 𝑥+9
?
?
?
What do you notice about the coefficient of the term in each case?
(𝑥+𝑎 )2=𝑥2+2𝑎𝑥+𝑎2?
Completing the Square – Starter
![Page 29: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/29.jpg)
Completing the square
Typical GCSE question:“Express in the form , where and are constants.”
(𝑥+3 )2−9?
![Page 30: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/30.jpg)
Completing the square
More examples:?
?
??
??
![Page 31: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/31.jpg)
Exercises
1
234
5
67
8
9
10
??
???
??
?
?
?
Express the following in the form
11 𝑥2+2𝑎𝑥+1= (𝑥+𝑎 )2−𝑎2+1?
![Page 32: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/32.jpg)
More complicated cases
Express the following in the form :
????
?
?
![Page 33: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/33.jpg)
Exercises
Put in the form or
1
234
5
6
7
????
??
?
![Page 34: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/34.jpg)
Proofs
Show that for any integer , is always even.
How many would we need to try before we’re convinced this is true? Is this a good approach?
![Page 35: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/35.jpg)
Proofs
Prove that the sum of three consecutive integers is a multiple of 3.
We need to ensure this works for any possible 3 consecutive numbers. What could we represent the first number as to keep things generic?
![Page 36: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/36.jpg)
Proofs
Prove that odd square numbers are always 1 more than a multiple of 4.
![Page 37: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/37.jpg)
How would you represent…
Any odd number: 2𝑛+1
Any even number: 2𝑛
Two consecutive odd numbers. 2𝑛+1 ,2𝑛+3
One less than a multiple of 3.
3𝑛−1
?
?
?
?
Two consecutive even numbers. 2𝑛 ,2𝑛+2?
![Page 38: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/38.jpg)
Proofs
Prove that the difference between the squares of two odd numbers is a multiple of 8.
![Page 39: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/39.jpg)
People in the left row work on this:
People in the middle row work on this:
People in in the right row work on this:
Example Problems
[June 2012] Prove that is a multiple of 8 for all positive integer values of .
[Nov 2012] (In the previous part of the question, you were asked to factorise , which is )
“ is a positive whole number. The expression can never be a prime number. Explain why.”
[March 2013] Prove algebraically that the difference between the squares of any two consecutive integers is equal to the sum of these two integers.
![Page 40: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/40.jpg)
Exercises
Edexcel GCSE Mathematics Textbook
Page 469 – Exercise 28EOdd numbered questions
![Page 41: GCSE: Quadratic Functions and Simplifying Rational Expressions](https://reader033.vdocuments.us/reader033/viewer/2022061612/568161ae550346895dd16e23/html5/thumbnails/41.jpg)
Even/Odd Proofs
Some proofs don’t need algebraic manipulation. They just require us to reason about when our number is odd and when our number is even.
Prove that is always odd for all integers .When is even: is . So is .
When is odd: is . So is .
Therefore is always odd.
?