Algebra formulas and theorems

Question 1

a^n - b^n = ?

Answer 1

(a - b)(a^n-1 + a^n-2b + … + ab^n-2 + b^n-1)

Question 2

a^n + b^n = ?

Answer 2

(a + b)(a^n-1 - a^n-2b + … - ab^n-2 + b^n-1)

Question 3

x^3+y^3+z^3-3xyz = ?

Answer 3

1/2 (x+y+z)((x-y)^2+(y-z)^2+(z-x)^2)

Question 4

x^3-y^3 = ?

Answer 4

(x-y)^3 - 3xy(x-y)

Question 5

x^3+y^3 = ?

Answer 5

(x+y)^3 - 3xy(x+y)

Question 6

(x+y+z)^3 = ?

Answer 6

x^3+y^3+z^3+3(x+y)(y+z)(z+x)

Question 7

(x+y+z)^2 = ?

Answer 7

x^2 + y^2 + z^2 + 2(xy + yz + zx)

Question 8

(x^2+y^2)(z^2+t&2) = ?

Answer 8

(xz+yt)^2 + (xt-yz)^2

Question 9

(x^2+y&2)(z^2+t^2) = ?

Answer 9

(xz-yt)^2 + (xt+yz)^2

Question 10

Simon’s favorite factoring trick: ab+ma+nb+k = ?

Answer 10

(a+n)(b+m)+k-mn

Question 11

Lemma: LOOK AT THE BOOK. I DON’T REALLY KNOW IT WELL

Answer 11

1.1.3.11

Question 12

Newton Sums

Answer 12

1.1.3.12

Question 13

De Moivre

Answer 13

1.1.4.1

Question 14

Euler’s formula

Answer 14

e^(ix) = cos(x) + i * sinx

Question 15

Roots of unity: z^n = 1

Answer 15

z = cos(2pik/n) + isin(2pik/n)

Question 16

HM GM AM QM inequality

Answer 16

1.1.5.1

Question 17

Sedrakyan’s inequality for squares

Answer 17

1.1.5.3

Question 18

Volume of a cone

Answer 18

pir^2x/3

Question 19

Area of cone

Answer 19

piry+pir^2

Question 20

Volume of a sphere

Answer 20

4/3 pir^3

Question 21

Area of a sphere

Answer 21

4pir^2

Question 22

Area of regular pentagon

Answer 22

sqrt(5(5+2sqrt(5)))a^2/4

Question 23

Area of regular hexagon

Answer 23

3sqrt(3)a^2/2

Question 24

Area of regular octagon

Answer 24

2(1+sqrt(2))a^2

Question 25

Area of regular decagon

Answer 25

5sqrt(5+2sqrt(5))a^2/2

Question 26

Area of regular dodecagon

Answer 26

3(2+sqrt(3))a^2

Question 27

Area of regualr polygon

Answer 27

na^2/4tan(180/n)

Question 28

sin(x+y)

Answer 28

sinx cosy + cosx siny

Question 29

sin(x-y)

Answer 29

sinx cosy - cosx siny

Question 30

cos(x+y)

Answer 30

cosx cosy + sinx siny

Question 31

cos(x-y)

Answer 31

cosx cosy - sinx siny

Question 32

tan(x+y)

Answer 32

(tan x + tan y)/(1 - tanx tany)

Question 33

tan(x-y)

Answer 33

(tan x - tan y)/(1 + tanx tany)

Question 34

sinx + siny

Answer 34

2sin((x+y)/2)cos((x-y)/2)

Question 35

sinx - siny

Answer 35

2sin((x-y)/2)cos((x+y)/2)

Question 36

cosx + cosy

Answer 36

2cos((x+y)/2)cos((x-y)/2)

Question 37

cosx - cosy

Answer 37

-2cos((x+y)/2)sin((x-y)/2)

Question 38

sinxsiny

Answer 38

1/2(cos(x-y) - cos(x+y))

Question 39

cosxcosy

Answer 39

1/2(cos(x-y) + cos(x+y))

Question 40

sinxcosy

Answer 40

1/2(sin(x-y) sin(x+y))

Question 41

sin2x

Answer 41

2sinxcosx

Question 42

cos2x

Answer 42

cos^2x-sin^2x

Question 43

tan2x

Answer 43

2tanx/(1-tan^2x)

Question 44

sin(x/2)

Answer 44

+/- sqrt((1-cosx)/2)

Question 45

cos(x/2)

Answer 45

+/- sqrt((1+cosx)/2)

Question 46

tan(x/2)

Answer 46

sinx/(1+cosx)