Chapter 7 - Algebraic Methods

Question 1

When simplifying an algebraic fraction what do you do?

Answer 1

You factorise the numerator and denominator where possible. You then cancel out the factors.

Question 2

What are the steps for polynomial division when dividing by (x +/- p) where p is a constant?

Answer 2

• Divide the first term by x
• Then multiply the new x term by the factor (x +/- p)
• Subtract that new term from the section above it.
• Write the answer of that below
• ‘Drop’ the next term (from the original equation) and put it next to your answer you just got.
• Divide the first term of that answer by x
• And repeat the steps until you get to the end.

Question 3

What are the 2 things that factor theorem states if f(x) is a polynomial?

Answer 3

• If f(p) = 0 then (x - p) is a factor of f(x).
• If (x - p) is a factor of f(x), then f(p) = 0.

Question 4

What is proof through deduction?

Answer 4

Starting from known facts or definitions, then using logical steps to reach the desired conclusion.

Question 5

When proving something mathematically, what 5 things must you do?

Answer 5

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 possibly cases.
• Write a statement of proof at the end of your working.

Question 6

What 3 things should you do when proving an identity?

Answer 6

You should:

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

Question 7

What is proof by exhaustion?

Answer 7

It means breaking the statement into smaller cases and solving each case separately. You must cover every possible case.

Question 8

What is proof by a counter-example?

Answer 8

Find one example that does not work for the statement. This only needs one example to disprove a statement.