Algebraic Techniques and Polynomials, Cheat Sheet of Mathematics

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Algebraic Techniques and Polynomials

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A LGEBRAIC TECHNIQUES

Recurring decimal^ to^ fractions

EXAMPLES

a 1. 329x =^1.^329 b 0.^ 215x^ = 0. 216 C 1.^0741 = 1.^0746

10x = 13. (^29) 100x = 21. (^6 1000) = 1074. I

• 990x = 1316 900x = 195 9000x (^) = 9967 1000s = 1329. (^29) x (^) = 13166 T 1000x = 216. (^6) x = 195 = (^13 10000) = 10741. / 9000 908 60

Index laws^ summary

au xam = a^ n^ +^ m Examples^ -^ Multiplication^ ,^ Division^ ,^ Power^ of^ a^ power^ Examples - Zero and negative indices

Example Fractional indices

(an)m = anm^ a

2 xy

a 4 Xb9^ a 2 =^3 -^3 =^2745 x7^ =^1 + O

am = an = am^ -^ n

2x xy = x - 5 xy^2 a^

• 4 = (^4) a (^) = 1 x^ - 5 xy4 (^3) x + 1 = (^) (x + 1 -^ I 2 275 = 327 = 3 a (^) = a 3 (2)" "in 3(a^ +^ b)^ = = a + b a - = (

at = n A

am = nam

Expansions

Binomial Expansions Examples

a PerfectSquare b Difference of 2 squares ( Cubics & fractional expansions

(2x -^ 3)2^ (2x^ -^ 7)(2x^ +^7 (x^ +^ 5)(2x^ +^ 7)(x^ -^ 9)^ (5x -^3 2x^ +^5 = 4x2 -^ 12x^ +^9 = 4x2^ -^49 2x^ -^ x^ - 118x -^315 2x + (^) jjx - xx (^) - = (^) Ex - 5. 9x - 3

Facto rising

a (^) common factors b (^) grouping )^ Difference of^ squares & Trinomials (^) e) (^) Perfect Squares f (^) quadratic formula

(x -^ 3)2^ +^ 5(^ -^ 3)^ -^ b^ =^ b)^ -^ 49)

Tr + 2 πr^ - 3r - 6 49x2^ -^4 monic^ m^2 -^ 5m^ +^6 49m2^ +^ 84m^ +^36

x2 - 6 + 9 + 5x -^1511 r v + 2 - 3 + 2 = 7)) - 27x + (^2) = (m - 1)(m + 5 (7m + (^) 6)2 x + (^) 3x - 1

x - - 6 = (r + 2)(πr - 3) non-monic 36a2-12a + / - 31(3)2 - 4(1)) - 1)

= (x^ -^ 3)(x^ +^2 36a2^ -^ 6a^ -^ Ga^ +^1 - 3 =T + 4

6a(ba - 1) - (6a - 1)^2

= (6a - 1)2 =^ -^31132

Algebraic Fraction

a Adding fractions b^ Subtracting Fractions^ C^ Multiplying Fractions^ d Dividing Fractions^ C^ 2x(5x)^ -^3

=^10

I^ x2^ -^9 x^ +^3

t^3 - 4

x 2 -^ 2x -^3 Y^ x2^ -^25 15 -

a -^ b a + b (^) x" - (^16) x - 4x - (^5) (x - 3)(x + 5 10 53333

a +^ b^ + a -^ b^29 3 - (2x - 8) x + 8 x y^102 -^6 (C^ +^ 3)^ =^ 10x(x2^ -^ 9)

= (a - b)(a +^ b)(a-^ b)(a+^ b)^ (x^ +^ 4)(x^ -^ 4)^ (x^ +^ 4)(x^ -^ 4)^ - (x - 3(x + (^1) (x + 5)(x - 5 = 5x(3x - y) (^) 2y 3 +^ 30x 242 -^6 -^ 18x^ = X X

(x - 5)(x +^ 1) (x - 3)(x + 5) 10 y 3x - Y 10x3 - 90

= I^ =^ / 24(+72x = 0

24x(x +^ 3)^ =^0

x = 0( =^ -^3

SurdS

a perfect squares b Adding 3 Subracting surds C Multiplying surds d) dividing surds e Binomial surds

(25 -^ - 3 2 + 2 -^52 +^424 +^9 103(2 - 212 728 i3 - 23 +^ 2ii(p +^ +^ p^ -^4

20 - 28 15 + 147 62 - 52 +^86 + 3 106 - 120 728 =^3 -^2 =^1 P^ -^ pp^ +^ 5 p-

7

I 28 = p -

= (^) 167- 2815 = 3 + 8062 PY+^ p^ -^1

f Rationalising denominators

52 +^1 3 + 2

X

3 -^2 3 + 2

Equations and^ inequalities

a Exponentials b Linear^ inequalities

ipm = 222m = 2 t^ ii24x^ +^1 = 8 24x +^1 = 23 - 26y -^37 =

2m = = m (^) = 4x^ +^1 =^ 3xx =^ -^12 so j