1 / 41

# GCSE: Quadratic Functions and Simplifying Rational Expressions - PowerPoint PPT Presentation

GCSE: Quadratic Functions and Simplifying Rational Expressions. Dr J Frost (jfrost[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?.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' GCSE: Quadratic Functions and Simplifying Rational Expressions' - najila

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### GCSE: Quadratic Functions and Simplifying Rational Expressions

Dr J Frost ([email protected])

Factorising means :

To turn an expression into a product of factors.

So what factors can we see here?

Year 8 Factorisation

Factorise

2x2 + 4xz

2x(x+2z)

Year 9 Factorisation

Factorise

x2 + 3x + 2

(x+1)(x+2)

A Level Factorisation

Factorise

2x3 + 3x2 – 11x – 6

(2x+1)(x-2)(x+3)

5 + 10x

x – 2xz

x2y – xy2

10xyz – 15x2y

xyz – 2x2yz2 + x2y2

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

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

11)

12)

?

?

?

?

?

?

?

?

?

?

?

?

?

It’s fine to factorise out an entire expression:

?

?

?

?

Edexcel GCSE Mathematics Textbook

Page 111 – Exercise 8D

Q1 (right column), Q2 (right column)

1

?

2

?

3

?

4

?

5

?

6

?

7

?

8

?

9

?

10

?

11

?

12

?

13

?

14

?

There’s a quick way to expand squared brackets involving two terms:

?

?

?

?

1. Factoring out a term

2.

?

?

4.

3. Difference of two squares

?

?

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

Which is ?

How does this suggest we can factorise say ?

?

Is there a good strategy for working out which numbers to use?

1

?

2

?

3

?

?

4

?

5

?

6

?

7

?

8

?

9

?

10

?

Firstly, what is the square root of:

?

?

?

?

?

Click to Start Bromanimation

?

?

?

?

?

?

Exercises:

?

1

?

2

?

3

?

4

?

5

?

6

?

7

?

8

?

9

?

10

?

Factorise using:

The ‘commando’ method*

b. Splitting the middle term

* Not official mathematical terminology.

?

?

?

?

?

1

?

2

?

3

4

?

5

?

‘Commando’ starts to become difficult from this question onwards.

6

?

?

7

?

8

?

9

?

10

11

?

Well Hardcore:

?

N

?

N

?

?

1

7

2

?

8

?

?

3

9

?

4

?

10

?

5

?

?

11

6

?

?

?

How did we identify the new denominator to use?

(Note: If you’ve added/subtracted fractions before using some ‘cross-multiplication’-esque method, unlearn it now, because it’s pants!)

The same principle can be applied to algebraic fractions.

?

?

?

?

?

“To learn the secret ways of algebra ninja, simplify fraction you must.”

?

?

?

?

?

?

1

?

8

?

2

?

9

?

3

?

10

?

4

11

?

5

?

?

6

?

7

Expand the following:

?

?

?

?

What do you notice about the coefficient of the term in each case?

Typical GCSE question:

“Express in the form , where and are constants.”

?

More examples:

?

?

?

?

?

?

Express the following in the form

1

?

?

2

3

?

4

?

?

5

6

?

7

?

?

?

11

8

?

9

?

10

Express the following in the form :

?

?

?

?

?

?

Put in the form or

?

1

?

2

?

3

4

?

?

5

?

6

?

7

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?

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?

Prove that odd square numbers are always 1 more than a multiple of 4.

Any odd number:

?

?

Any even number:

Two consecutive odd numbers.

?

Two consecutive even numbers.

?

One less than a multiple of 3.

?

Prove that the difference between the squares of two odd numbers is a multiple of 8.

People in in the right row work on this:

People in the left row work on this:

People in the middle row work on this:

[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.

[June 2012] Prove that is a multiple of 8 for all positive integer values of .

Edexcel GCSE Mathematics Textbook

Page 469 – Exercise 28E

Odd numbered questions

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.

?