Lista de Exercícios - Pré-Cálculo - B

Lista de Exerc´ıcios – Pr´e-C´alculo – B Prof. Carlos Campani 16 de fevereiro de 2015 1. Efetue os seguintes produtos: (l) (5x3 − 1)2

(a) (x − 1)(x + 5) (b) (2x + 3)(4x + 1)

(m) (x2 − 2x + 3)(x + 4)

(c) (3x − 5)(x + 2)

(n) (x2 + 3x − 2)(x − 3)

(d) (2x − 3)(2x + 3)

(o) (x2 + x − 3)(x2 + x + 1)

(e) (3x − y)(3x + y)

(p) (2x2 − 3x + 1)(x2 − x + 2) √ √ (q) (x − 2)(x + 2)

(f) (3 − 5x)2 (g) (3x + 4y)2 (i) (2u − v)3

(r) (x1/2 − y 1/2 )(x1/2 + y 1/2 ) √ √ √ √ (s) ( u + v)( u − v) √ √ (t) (x2 − 3)(x2 + 3)

(j) (u + 3v)3

(u) (x − 2)(x2 + 2x + 4)

(k) (2x3 − 3y)(2x3 + 3y)

(v) (x + 1)(x2 − x + 1)

(h) (x − 1)3

2. Fatore cada express˜ao alg´ebrica: (a) xy − x

(g) x3 − 2x2 − 3x

(b) 100 − x2 y 2

(h)

(c) ax2 − ay 2

(i) x y − 2mxy 5

(d) 25x3 − 16x

(j) x2 − y − y 2 + x

(e) m3 p − p3 m

(k) 4y 6 + 4y 5 + y + 1

(f) 3x2 y − 6xy 3 + 9x2 y 2

(l) 2a3 + 6ax − 3a2 b − 9bx 1

1 4 2 xy 2 3 2

+ 41 x2 y 4

(m) x2 y + 2xy 2 − 2xy − 4y 2

(p) m4 x2 + 4m3 xy + 4y 2 m2

(n) 3x2 y 2 − 12xy + 12

(q) 9a2 x2 − 6ab3 x + b6

(o) y 4 − 6mxy 2 + 9m2 x2

(r)

1 2 x 9

− 16 xy +

1 2 y 16

3. Simplifique cada express˜ao alg´ebrica: (a)

a−5 a2 −25

(g)

xy+y+5x+5 3y+15

(b)

x2 −y 2 xy+y 2

(h)

x2 −y 2 x2 −2xy+y 2

(c)

a−2x 2bx−ab

(i)

x2 −5x+6 x2 −9

(d)

20x3 y 2 z 4 15x6 y 6 z

(j)

x4 −1 x4 +2x2 +1

(e)

x2 −4xy+4y 2 x2 −4y 2

(f)

a2 +2ac+c2 a2 −c2

(k)

x2 −x−6 2x2 +4x

(l)

x2 +4x−5 x2 −2x+1

4. Simplifique.

(c)

3 x2 −1 . 9 x−1 x+3 14 . 7 2x+6 x+3 1−x . x−1 x2 −9

(d)

18x2 −3x 12y 2 . 6x−1 3xy

(e)

x3 −1 . 4x 2x2 x2 +x+1

(f)

y 3 +2y 2 +4y y 2 −4 . y3 −8 y 3 +2y 2

(g)

2y 2 +9y−5 y−5 . 2y2 −y y 2 −25

(h)

y 2 +8y+16 3y 2 +2y . y+4 3y 2 −y−2 1 ÷ 41 2x 4x ÷ 8y y x

(a) (b)

(i) (j) (k) (l)

x2 −3x 14y 7x−7y 4y

÷ ÷

2xy 3y 2 14x−14y 3y

2

(m)

2x2 y (x−3)2 8xy x−3

(n)

x2 −y 2 2xy y 2 −x2 4x2 y

(o)

2x+1 x+5

(p)

3 x−2

(q)

3 x2 +3x

(r)

5 x2 +x−6

−

2 x−2

(s)

4x2 −7xy 3x2

+

8y 2 −3x 6x

(t)

5 2x+2

−

7 3x−3

+

1 6x−6

(u)

x+1 2x−2

−

x−1 2x+2

+

4x x2 −1

(v)

4t2 t2 −s2

−

t−s t+s

3 x+5

−

+

x+1 x−2

−

1 x

−

6 x2 −9

+

+

4 x2 −4

−

t+s t−s

5 12

5. Resolva a equa¸c˜ao. (a) 3x = 24

(h) 4(y − 2) = 5y

(b) 4x = −16

(i)

(c) 3t − 4 = 8

(j)

(d) 2t − 9 = 3

(k)

(e) 2x − 3 = 4x − 5

(l)

1 x 2 2 x 3 1 x 2 1 x 3

= = + +

7 8 4 5 1 3 1 4

=1 =1

(f) 4 − 2x = 3x − 6

(m) 2(3 − 4z) − 5(2z + 3) = z − 17

(g) 4 − 3y = 2(y + 4)

(n) 3(5z − 3) − 4(2z + 1) = 5z − 2

6. Resolva a equa¸c˜ao. (a) x2 − x − 6 = 0

(h) x2 + 3x + 2 = 0

(b) x2 + 2 = 3x

(i) x2 − x − 20 = 0

(c) 3x2 + 12x + 12 = 0

(j) 2x2 + 5x − 3 = 0

(d) 3x2 + 3x − 6 = 0

(k) 4x2 − 8x + 3 = 0

(e) x2 + 4x − 5 = 0

(l) x2 − 8x = −15

(f) 3x2 − x + 2 = 0

(m) x(3x − 7) = 6

(g) 2x2 + 4x = 16

(n) x(3x + 11) = 20

7. Fatore. (a) 2yx2 − y + xy

(c) (x + 1)3 − y 3

(b) 2yx2 + 2y − 4xy

(d) z 3 − (x + 2)3

8. Resolva as inequa¸co˜es abaixo e apresente a solu¸ca˜o em nota¸c˜ao de intervalos: (e) x(2x + 3) ≥ 5

(a) 2x + 5 < 3x − 7 (b) 3 ≤

2x−3 5

(f) |x + 3| < 0, 01

3

(h) |6 − 5x| ≤ 3 3

(i) |3x − 7| ≥ 5

(r) x − 7 ≥ −5 ou x − 7 ≤ −6

(j) | − 11 − 7x| > 6

(s) x < 6x − 10 ou x ≥ 2x + 5

(k) −5 ≤ 3x + 4 < 7

(t) 2x − 1 > 1 ou x + 3 < 4

(l) |6x − 7| > 10

(u) 1 ≤ −2x + 1 < 3

(m) 0 < 3x + 1 ≤ 4x − 6

(v) x + 3 < 6x + 10

(n) |5 − 2x| ≥ 7 (o) −6 < 3x + 3 ≤ 3

(w) |2x − 3| > 4

(p) |x − 4| ≤ 16

(x) 2 < 5x + 3 ≤ 8x − 12

(q) 1 < x − 2 < 6 − x

(y) |2x − 3| ≤ 5

9. Resolva a inequa¸c˜ao e represente a solu¸ca˜o graficamente na reta real. (a) (b)

5x+7 4 3x−2 5

(c) 4 ≥

≤ −3

(g)

> −1

(h)

2y−5 3 3y−1 4

≥ −2

(i)

> −1

(j)

(e) 0 ≤ 2z + 5 < 8

(k)

(f) −6 < 5t − 1 < 0

(l)

(d) 1 >

x−5 + 3−2x < −2 4 3 3−x + 5x−2 < −1 2 3 2y−3 3y−1 + 5 3, 6

(f) |3 − 2x| + 2 > 5

(c) |x − 3| < 2

(g) x+2 ≥3 3

(d) |x + 3| ≤ 5

(h) x−5 ≤6 4









11. Resolva. (a) |x + 1| + |x − 3| < 5 (b)

3x−2 5−x

(d) (e)

x+2 2x+1 >3 x−1 2 x +2x−6 ≥ x+2

0 −1