P2 §1. Algebraic methods

Pure mathematics Year 2

Here's a short introduction to Pure maths 2 Chapter 1 Algebraic methods.

 

1.1 Proof by contradition: Use proof by contradiction to prove true statements.

1.2 Algebraic fractions: Multiply and divide two or more algebraic fractions; Add or subtract two or more algebraic fractions.

1.3 Partial fractions: Convert an expression with linear factors in the denominator into partial fractions.

1.4 Repeated factors: Convert an expression with repeated linear factors in the denominator into partial fractions.

1.5 Algebraic division: Divide algebraic expressions; Convert an improper fraction into partial fraction form.

1.6 Mixed exercise for chapter 1

1.5 Review exercise for chapter 1

 

Prior knowledge check

 

Q1. [P1 §1.3 Factorising] Factorise each polynomial:

(a) $ x^2 - 6x + 5 $

(b) $ x^2 - 16 $

(c) $ 9x^2 - 25 $

 

Answers:

(a) $ (x-1)(x-5) $

(b) $ (x+4)(x-4) $

(c) $ (3x+5)(3x-5) $

 

Q2. [P1 §7.1 Algberaic fractions] Simplify fully the following algebraic fractions.

(a) $ \frac{ x^2 - 9 }{ x^2 + 9x + 18 } $

(b) $ \frac{ 2x^2 + 5x - 12 }{ 6x^2 - 7x - 3 } $

(c) $ \frac{ x^2 - x - 30 }{ -x^2 + 3x + 18 } $

 

Answers:

(a) $ \frac{ x-3 }{ x+6 } $

(b) $ \frac{ x+4 }{ 3x+1 } $

(c) $ - \frac{ x+5 }{ x+3 } $

 

Q3. [P1 §7.5 Methods of proof] For any integers $n$ and $m$, decide whether the following will always be odd, always be even, or could be either.

(a) $ 8n $

(b) $ n-m $

(c) $ 3m $

(d) $ 2n-5 $

 

Answers:

(a) even

(b) either

(c) either

(d) odd

 

Comment: We can use proof by contradiction to prove that there is an infinite number of prime numbers. (See P2 §1.1 Proof by contradiction Example 4.) Very large prime numbers are used to encode chip and pin transactions.