Pure Mathematics - Algebraic Methods

Question 1

What is a polynomial?

Answer 1

A polynomial is a finite expression with positive whole number indices.

Question 2

What is a mathematical proof?

Answer 2

• A proof is a logical and structured argument to show that a mathematical statement (or conjecture) is always true.
• A mathematical proof usually starts with previously established mathematical facts (or theorems) and then works through a series of logical steps. The final step in a proof is a statement of what has been proven.

Question 3

What are the methods of proof?

Answer 3

• deduction
• exhaustion
• counter-example

Question 4

What is deduction?

Answer 4

You can prove a mathematical statement is true by deduction. This means starting from known facts or definitions, then using logical steps to reach the desired conclusion.

Question 5

What must you do in a mathematical proof?

Answer 5

In a mathematical proof, you must:

• State any information or assumptions you are using.
• Show every step of your proof clearly.
• Make sure that every step follows logically from the previous step.
• Make sure you have covered all possible cases.
• Write a statement of proof at the end of your working.

Question 6

How do you prove an identity?

Answer 6

To prove an identity, you should:

• Start with the expression on one side of the identity (either side).
• Manipulate that expression algebraically until it matches the other side.
• Show every step of your algebraic working.

Question 7

What is exhaustion?

Answer 7

You can prove a mathematical statement is true by exhaustion. This means breaking the statement into smaller cases and proving each case separately.

Question 8

What is a counter-example?

Answer 8

You can prove a mathematical statement is not true by a counter-example. A counter-example is one example that does not work for the statement. You do not need to give more than one example, as one is sufficient to disprove a statement.