P2 Chapter 1 :: Algebraic Methods
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Chapter Overview
Apart from the initial portion on proof, this chapter concerns manipulation of algebraic fractions.
1:: Proof By Contradiction
3:: Express a fraction using partial fractions.
4:: Divide algebraic expressions, and convert an improper fraction into partial fraction form.
1 :: Proof By Contradiction
🖉 To prove a statement is true by contradiction:
Prove that there is no greatest odd integer.
? Assumption
? Show contradiction
? Conclusion
How to structure/word proof:
Negating the original statement
The first part of a proof by contradiction requires you to negate the original statement.
What is the negation of each of these statements? (Click to choose)
“There are infinitely many prime numbers.”
“There are infinitely many non-prime (i.e. composite) numbers.”
“There are finitely many prime numbers.”
“There are finitely many non-composite numbers.”
“All Popes are Catholic.”
“No Popes are Catholic.”
“There exists a Pope who is not Catholic.”
“Dr Frost is the Pope.”
“If it is raining, my garden is wet.”
“It is not raining and my garden is dry.”
“It is raining and my garden is not wet.”
“It is not raining and my garden is wet.”
Comments: The negation of “all are” is not “none are”. So the negation of “everyone likes green” wouldn’t be “no one likes green”, but: “not everyone likes green”. Do not confuse a ‘negation’ with the ‘opposite’.
Comments: If you have a conditional statement like “If A then B”, then the negation is “A and not B”, i.e. the condition is true, but the conclusion is false/negated.
More Examples
Remember the negation of “if A then B” is “A and not B”.
? Assumption
? Show contradiction
? Conclusion
More Examples
? Assumption
? Show contradiction
? Conclusion
More Examples
Prove by contradiction that there are infinitely many prime numbers.
This proof is courtesy of Euclid, and is one of the earliest known proofs.
? Assumption
? Show contradiction
? Conclusion
Exercise 1A
Pearson Pure Mathematics Year 2/AS
Pages 4-5
Extension
Solution to Extension Problem
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Solution to Extension Problem
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2a :: Multiplying/Dividing Algebraic Fractions
As your saw at GCSE level, multiplying algebraic fractions is no different to multiply numeric fractions.
You may however need to cancel common factors, by factorising where possible.
To divide by a fraction, multiply by the reciprocal of the second fraction.
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Test Your Understanding
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Common student “Crime against Mathematics”:
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Exercise 1B
Pearson Pure Mathematics Year 2/AS
Pages 6-7
2b :: Adding/Subtracting Algebraic Fractions
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Test Your Understanding
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Exercise 1C
Pearson Pure Mathematics Year 2/AS
Page 8
Partial Fractions
If the denominator is a product of a linear terms, it can be split into the sum of ‘partial fractions’, where each denominator is a single linear term.
Method 1: Substitution
Method 2: Comparing Coefficients
We don’t like fractions, so multiply through by denominator of LHS.
See note below.
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Further Example
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Test Your Understanding
C4 June 2005 Q3a
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Exercise 1D
Pearson Pure Mathematics Year 2/AS
Page 11
Repeated linear factors
Q
The problem is resolved by having the factor both squared and non-squared (explanation of why we do this at the end of these slides).
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Test Your Understanding
C4 June 2011 Q1
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Exercise 1E
Pearson Pure Mathematics Year 2/AS
Page 13
Dealing with Improper Fractions
In Pure Year 1, we saw that the ‘degree’ of a polynomial is the highest power, e.g. a quadratic has degree 2.
An algebraic fraction is improper if the degree of the numerator is at least the degree of the denominator.
A partial fraction is still improper if the degree is the same top and bottom.
Reducing to Quotient and Remainder
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Quotient
Edexcel C4 June 2013 Q1
Test Your Understanding
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Exercise 1F
Pearson Pure Mathematics Year 2/AS
Pages 16-17
Dealing with Improper Fractions
Q
Method 1: Algebraic Division
Method 2: Using One Identity
Bropinion: I personally think the second method is easier. And mark schemes present it as “Method 1” – implying more standard!
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Test Your Understanding
C4 Jan 2013 Q3
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Exercise 1G
Pearson Pure Mathematics Year 2/AS
Page 18
Informal proof of method for repeated factors
If a factor is repeated, why do have a partial fraction with the squared and one without?
When we split into partial fractions, we want each fraction to be non-top-heavy algebraic fractions – recall this means that the ‘order’ of the numerator has to be less than the order of the denominator. We assume the most generic non-top-heavy fraction possible, i.e. where the order of the numerator is one less than the denominator…
order 2
order 1
Split the fraction.
Informal proof of method for repeated factors
Split the fraction as before.