Algebra

Question 1

When to use –> and for an example see 2008 4ii

Answer 1

Spot when we have an identity rather than an equality so compare coefficients

Question 2

d = a^x * b^y * c^z therefore d has how many factors:

Answer 2

(x+1)(y+1)(z+1)

Question 3

(A+1)(B+1) =

Answer 3

AB + A + B + 1

Question 4

(A+B)(A+B) =

Answer 4

A^2 + 2AB + B^2

Question 5

(A-B)(B-C)(C-A) =

Answer 5

A^2 + B^2 + C^2 - AB - AC - CA

Question 6

Arithmetic mean > geometric mean

Answer 6

0.5(x+y) > root(xy)

Question 7

k/k-k/k-k/k-k/k… therefore

Answer 7

x = k/k-x

Question 8

Finding solutions

Answer 8

factorise, reason about the graph, consider the discriminant

Question 9

99^2 =

Answer 9

100-1)^2 = 100^ - 200 +1

Question 10

Divide by x^2 +3x + 2 means

Answer 10

F(-2) = F(-1)

Question 11

Identity allows us to

Answer 11

compare coefficients rather than equality

Question 12

With limits, constants…

Answer 12

become inconsequential

Question 13

a^2 + b^2 = 1

Answer 13

a is greatest when b = 0

Question 14

For a quadratic to have maximum value,

Answer 14

the x^2 coeffecient must be negative

Question 15

a^x > cb^y –>

Answer 15

find counterexamples, use logs

Question 16

Complete the square

Answer 16

maximum when leading coefficient is -ve

Question 17

Use algebra to show things…:

Answer 17

(x-y)^2

Question 18

Dividing by algebra

Answer 18

think of factor theorem or subbing in values

Question 19

When finding powers, check if coefficients cancel

Answer 19

Compare coefficients

Question 20

x^3 - x^2 - x + 1 = 0 –>

Answer 20

x^2(x-1)-1(x-1) = (x^2 - 1)(x-1)

Question 21

Max/min –>

Answer 21

complete the square

Question 22

a^4 - a^2 =

Answer 22

(a+1)(a-1)(a^2 + 1)

Question 23

x^3 + 6yx^2 + 12xy^2 + 8y^3 =

Answer 23

(x+2y)^3

Question 24

Don’t have fractions in equations of lines i.e. 1/a as

Answer 24

prevents a from equaling 0

Question 25

For factorizing, if coefficient = 2

Answer 25

try +/- 0.5

Question 26

(a+b)^6 =

Answer 26

a^6 + 6ba^5 + (6,2)a^4b^2 + (6,3)a^3b^3 + … + b^6

Question 27

SinXcosX <= 0.5 use

Answer 27

(sinX - cosX)^2 > 0

Question 28

a^2 - a^1.5 - 8 = 0

Answer 28

solutions

Question 29

Expand (ax-by)^n

using choose function and pascals triangle

Answer 29

C(n,0)x^0(-by)^n
\+
C(n,1)x^1(-by)^n-1
\+
...
\+
C(n,n)x^n(-by)^0

Pascals triangle

Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1
....
1's on the outside, add both numbers above to get number below

therefore (ax-by)^n would follow with decreasing power of ax from n to 0, increasing power of -by from 0 to n, and coefficients reading left to right of the nth row of pascals triangle

(ax-by)^4, coefficients would b 1, 4, 6, 4 ,1

Question 30

C(n,r)

Answer 30

n!/r!(n-r)!

Question 31

X^2+1 is a factor of p(x)

Answer 31

sub x^2 = -1 into it then p(x^2=-1) = 0

Question 32

Base cases: a and b are positive integers

Answer 32

let a, b =1

Question 33

Expand (x-1)^3

Answer 33

x^3 - 3x^2 + 3x - 1

Question 34

2x +3y < a

Answer 34

Form y = mc + c