1. Algebraic Methods

Question 1

2 part partial fractions

Answer 1

Multiply A,B by the opposite denominator and find A,B

Question 2

Partial fraction methods

Answer 2

1. Substitute the x to make brackets 0 to allow you to find A,B
   OR
2. Equate each coefficient equally and use simultaneous equations

Question 3

2 part repeated factors

Answer 3

A B
——— + ————
factor factor^2

Question 4

3 part repeated factors

Answer 4

Multiply A,B,C each by what their respective denominators must be to make it the same as the overall denominator

Question 5

Improper fractions

Answer 5

The degree of the numerator is higher than the degree of the denominator

Question 6

Reducing to quotient and remainder

Answer 6

Divide the numerator by the denominator for the quotient

Answer = quotient + remainder/denominator

Question 7

Quotient and partial fractions

Answer 7

Divide for the quotient and convert the remainder into partial fractions

Question 8

Proof by Contradiction

Answer 8

1. Start by assuming the original statement is false
2. Use logical steps to show that this contradiction leads to something impossible
3. You can assume that your assumption was incorrect and the original value true

Question 9

Proving root 2 is irrational by contradiction

Answer 9

Start by assuming root 2 is rational and can be written as a/b, where a,b are integers and a/b is in its simplest form
Show that a,b must be even as both of their squares are even (show a is even then write b^2 in terms of a^2)
Therefore a/b must not be in its simplest form - contradiction

Question 10

Proving there are infinitely many primes

Answer 10

Start by assuming there are a finite number of primes, p1,p2,p3…pn
Let P = p1 x p2 x p3 x … x pn + 1
If P is prime it is bigger than any on the list
If P is not prime it must have a prime factor which is a factor of p1 x p2 x … x pn and 1, contradiction