Chapter 1 Algebraic Methods (Year 2)

Question 1

How do you prove a statement by contradiction?

Answer 1

• start by assuming the statement isn’t true
• use logical steps to show that this assumption leads to something impossible (either a contradiction of the assumption or a contradiction of a fact you know to be true)

Question 2

What is the difference between a rational number and an irrational number?

Answer 2

• a rational number can be written as a/b where a and b are integers
• an irrational number cannot be written as a/b

Question 3

How do you multiply fractions?

Answer 3

• cancel any common factors
• then multiply the numerators and multiply the denominators

Question 4

How do you divide two fractions?

Answer 4

Multiply the first fraction by the reciprocal of the second fraction

Question 5

How do you add or subtract two fractions?

Answer 5

Find a common denominator

Question 6

What happens when you split a fraction into partial fractions?

Answer 6

A single fraction with 2 distinct linear factors in the denominator can be split into two separate fractions with linear denominators

Question 7

When can the method of partial fractions be used?

Answer 7

When there are two or more distinct linear factors in the denominator

Question 8

How can a single fraction with a repeated linear factors in the denominator be split?

Answer 8

It can be split into two or more separate fractions, with the denominator of one being the repeated factor and the denominator of another being the repeated factor squared (e.g. A/x+1 + B/(x+1)^2)

Question 9

What is an improper algebraic fraction?

Answer 9

One whose numerator has a degree equal or larger than the denominator

Question 10

Can you express an improper algebraic fraction as a partial fraction?

Answer 10

Yes, if you convert the improper fraction into a mixed fraction first