Algebraic Methods (P1.7 & 2.1)

Question 1

how can algebraic fractions be simplified?

Answer 1

factorise the numerator & denominator where possible then cancel common factors

Question 2

define polynomial

Answer 2

a finite expression with positive whole number indices

Question 3

how can polynomials be divided?

Answer 3

use long division to divide a polynomial by (x +/- p) where p is a constant

Question 4

polynomial long division practice

Question 5

describe the factor theorem

Answer 5

if f(x) is a polynomial, then:
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 6

describe the steps of general mathematical proof

Answer 6

start with known facts/theorems
state information or assumptions you use
show every step of the proof clearly
ensure the steps follow logically from the previous step
make sure all possible cases are covered
end with a statement of proof

Question 7

how can a statement be proven by deduction?

Answer 7

start from known facts/definitions
use logical steps to reach the desired conclusion
e.g. using 2p, 2p + 1 etc.

Question 8

how can identities be proven?

Answer 8

start with the expression on one side of the identity
manipulate the expression algebraically until it matches the other side
show every step of working

Question 9

prove factor theorem

Answer 9

see pg 147 P1 textbook

Question 10

describe proof by exhaustion

Answer 10

breaking the statement into smaller cases & proving each case separately e.g. pg 150 P1 textbook
better suited to a small number of results

Question 11

describe proof by counter-example

Answer 11

used to prove a statement is not true
give one example that does not work for the statement

Question 12

how can a statement be proven by contradiction?

Answer 12

first assume the statement is not true i.e. assume the opposite is true
then use logical steps to show this assumption leads to something impossible (contradiction of the assumption or something true)
then conclude the assumption is incorrect so original statement is true

Question 13

how can a rational number be written?

Answer 13

a/b where a & b are integers
an irrational number cannot be expressed in this form

Question 14

prove by contradiction that √2 is an irrational number

Answer 14

see pg 3 P2 textbook

Question 15

prove by contradiction that there are infinitely many prime #s

Answer 15

see pg 3 P2 textbook

Question 16

how do you multiply, divide, add & subtract fractions?

Answer 16

multiply: cancel common factors, multiply the numerators & multiply the denominators
divide: multiply the 1st fraction by the reciprocal of the 2nd fraction
to add/subtract: find common denominator

Question 17

how is a single fraction split into partial fractions?

Answer 17

2 (or more) distinct linear factors:
multiple both sides by denominator
substitute values of x to eliminate A, B (or C)
equate coefficients of the same degree of x

repeated linear factor:
A/factor 1 + B/ factor 2 not repeated + C/factor 2 repeated

Question 18

what is an improper fraction?

Answer 18

fraction whose numerator has a degree equal to or greater than the denominator

Question 19

how is an improper fraction converted into a mixed fraction?

Answer 19

an improper fraction must be converted into mixed fraction before expressing it in partial fractions

algebraic long division
inspection: relationship F(x) = Q(x) x divisor + remainder