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GCSE: Quadratic Functions and Simplifying Rational ExpressionsPowerPoint Presentation

GCSE: Quadratic Functions and Simplifying Rational Expressions

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### GCSE: Quadratic Functions and Simplifying Rational Expressions

Dr J Frost ([email protected])

Last modified: 25th August 2013

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)

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

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Edexcel GCSE Mathematics Textbook

Page 111 – Exercise 8D

Q1 (right column), Q2 (right column)

Faster expansion of squared brackets

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

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Four different types of factorisation

1. Factoring out a term

2.

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3. Difference of two squares

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

Which is ?

How does this suggest we can factorise say ?

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

Click to Start Bromanimation

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Factorise using:

The ‘commando’ method*

b. Splitting the middle term

* Not official mathematical terminology.

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

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Well Hardcore:

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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 Wall of Algebraic Fraction Destiny

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

Completing the Square – Starter

Expand the following:

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

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:

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Any even number:

Two consecutive odd numbers.

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Two consecutive even numbers.

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One less than a multiple of 3.

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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 .

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.

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