Chapter 7 Algebraic Methods

Question 1

how do you simplify an algebraic fraction using division?

Answer 1

• factorise the numerator and denominator
• cancel common factors

Question 2

how do you divide a polynominal by (x ± p) where p is a constant?

Answer 2

using long division

Question 3

what is the factor theorem?

Answer 3

if f(x) is a polynominal then:
- if f(p) = 0, then (x - p) is a factor
- if (x - p) is a factor, then f(p) = 0

Question 4

how can you use the factor theorem to quickly factorise a cubic function, g(x)?

Answer 4

• substitute values into the function until you find a value, p, such that g(p) = 0
• divide the function by (x - p)
• write g(x) as (x - p)(ax^2 + bx + c)
• factorise (ax^2 + bx + c) if possible to write g(x) as a product of 3 linear factors

Question 5

what must you do in mathematical proof?

Answer 5

• state any information or assumptions youre using
• show every step of your proof clearly
• make sure 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

• 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 working

Question 7

what does proving by deduction mean?

Answer 7

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

Question 8

what does proving by exhaustion mean?

Answer 8

breaking the statement into smaller cases and proving each case separately
this is better suited to a small number of results

Question 9

what does proving by counter-example mean?

Answer 9

you can prove a mathematical statement is not true by counter-example
a counter-example is one example that doesn’t work for the statement