1. Determine a raiz das equações.

a) 8-x = 2+x
8-2 = x²

b) 3 (x-6) = 5x
3x-18 = 5x
-18 = 2x
x = -18/2
x = -9

c) 2/3x + 1/4 x = 6
2/3 x + x/4 = 6/1
8x + 3x/12 = 72
11x = 72
x = 72/11

d) 2x-1/10 - 2 = 1/5 - 1+x/4
4x - 2 - 40 = 4 . 5 (1+x)
4x - 2 - 40 = 4 - 5-5x
4x + 5x = 4 - 5 + 2 + 40
9x = 41
x = 41/9

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2. Calcule as expressões usando produtos notáveis.

a) (3x+2)²
9x² + 12x + 4

b) (4x+3)³
(4x)³ + 3 (4x)² . 3 + 3.4x. (3)² + (3)³
64x³ + 3.16x² . 3 + 3.4x . 9 + 27
64x³ + 144x² + 108x + 27

c) (2x-6)²
(2x)² - 2 . 2x . - 6 + 6²
4x² + 24x + 36

d) (x-3)³
x² - 3 . x² . (-3) + 3 . x . (-3)² - (-3)³
x³ + 9x² + 27x + 27


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Raiz

Raiz de 529 = 23

Raiz de 4096 = 64

Raiz de 6084 = 78

Raiz de 9801 = 99

Raiz de 1225 = 35

Raiz de 729 = 27

Raiz de 4624 = 68

Raiz de 64516 = 254

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Fatorial

4! = 4.3.2.1 = 24

7! = 7.6.5.4.3.2.1 = 5040

3!.2! = 3.2.1 . 2.1 = 6.2 = 12

5!+2! = 5.4.3.2.1 + 2.1 = 120 + 20 = 122

6!+3!-2! = 6.5.4.3.2.1 + 3.2.1 - 2.1/5.4.3.2.1 = 720 + 6.2

4!-2!-0!/1! = 4.3.2.1 - 2.1 - 1/1 = 24 - 2 - 1/1 = 21

12!/9! = 12.11.10.9!/9! = 1320

7!.4!/10! = 7! 4.3.2.1/10.9.8.7! = 24/720 = 1/30

105!/104! = 105.104/104 = 105

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n!/(n-1) = n

n!/(n-2) = n² - n

n!(n+2) = n² + n

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10!+8!/8! = 10.9.8! + 8!/8! = 90 + 1 = 91

100!/99! - 99!/98! = 100 - 99!/99 - 99 . 98!/98! = 100 - 99 = 1

(m+2)/m = (m+2) (m+1) m!/m! = m² + 3m + 2

(2n+1)!/(2n-1)! = (2n + 1) 2n (2n - 1)/(2n - 1)! = 4n² + 2n

(1/n! - 1/(n+1)) . n! = n!/n! = n!/(n+1)! = 1 - n! / (n+1)! = 1 - n (n+1)!/(n+1) = 1 - n

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Equações Fatoriais

a) x! = 15 (x-1)!

x (x-1)!/(x-1) = 15

x = 15

b) (n-2)! = 2 (n-4)!

(n-2)!/(n-4)! = 2

n² - 3n - 2n + 6 = 2

n² - 5n + 4 = 0

Δ = -5² - 4 . 1 . 4

Δ = 25 - 16

Δ = 9

5 +- √9/2 = 5 + 3/2 = 8/2 = 4

              = 2/2 = 1

c) (x+1)!/(x-1) = 56

(x+1) x = 56

x² + x - 56 = 0

Δ = b² - 4. a . c

Δ = 1² - 4 . 1² . 56

Δ = 2 - 4 . 2 . 56

Δ = 2 . 2 . 56

Δ = 4. 56

Δ = 225

-1 +- √225/2.2 = -1 +- √15/4 =  32

                                         =  -7

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Binômio de Newton

a) (8/5) = 8! / 5!(8-5!) = 8! / 5!3! = 8.7.6.5! / 5! 3.2.1 = 336/6 = 56

b) (6/0) = 6! / 0!(6-0) = 6! / 0!6! = 6.5.4.3.2.1.0 / 0! 6.5.4.3 = 0

c) (7/2) = 7! / 2!(7-2) = 7! / 2! 5! = 7.6.5.4.3.2 / 2! 5.4.3.2 = 2520/60 = 42

d) (3!)!/3! = 3.2.1! / 3 = 6! / 3! = 6.5.4.3! / 3! = 120

e) (5/3) = 5! / 3! (3-5) = 5.4.3! / 3! 2! = 20 / 2 = 10

f) (4/2) = 4! / 2! (4-2) = 4.3.2 / 2! 2! = 12 / 2 = 6

g) (25/23) = 25! / 23! (25-23)! = 25 . 24 . 23 / 23! 2! = 600 / 2 = 300

h) (100/99) = 100! / 99! (100 - 99)! = 100 . 99 / 99! . 1! = 100