c) d)
m2x 2 x 5 x ? (m2 2 1) 5 5 x ? (m 1 1) (m 2 1) 5a2 1 30ab 1 45b2 5 5 5 (a2 1 6ab 1 9b2) 5 5 ? (a 1 3b)2
14 – Cálculo do m.m.c. de polinômios
e) x3y 2 xy3 5 xy(x2 2 y2) 5 5 xy ? (x 1 y) (x 2 y) f) m8 2 n8 5 (m4 1 n4) (m4 2 n4) 5 5 (m4 1 n4) (m2 1 n2) (m2 2 n2) 5 5 (m4 1 n4) (m2 1 n2) (m 1 n) (m 2 n) g)
Exercícios, páginas 114 e 115. 1. a) m.m.c. (54, 72) 5 23 ? 33 5 216 54 5 21 ? 33 72 5 23 ? 32
x3 2 xy2 1 x2y 2 y3 5 5 x (x2 2 y2) 1 y (x2 2 y2) 5 5 (x2 2 y2) (x 1 y) 5 5 (x 1 y) (x 2 y) (x 1 y) 5 5 (x 1 y)2 (x 2 y)
b)
m.m.c. (200, 100, 80) 5 24 ? 52 5 400 200 5 23 ? 52 100 5 22 ? 52 80 5 24 ? 5
c) m.m.c. (42, 63, 105) 5 2 ? 32 ? 5 ? 7 5 630 42 5 2 ? 3 ? 7 63 5 32 ? 7 1 105 1 1 51 3 ? 5 ? 7 p2 1 2 p2 5 1 1 p2 1 1 p 1 2 p 2d) m.m.c. 2 (18, 4 4 24, 36, 72) 5 23 ? 32 5 72 1 18 5 2 ? 32 p 1 2 p 2 24 5 23 ? 3
h) a4 2 ax3 5 a (a3 2 x3) 5 5 a (a 2 x) (a2 1 ax 1 x2) 1 4 1 p 5 1 1 16 4 1 1 1 1 p4 5 1 1 p2 1 2 p 2 5 1 1 p2 1 1 4 4 4 2 i) 1 2
2 36 5 22 ? 32 4 2 4 4 4 2 y 1 y 5 y y2 1 y 1 5 y y 1 72 5 23 ? 32 3 9 3 9 3 2 4 4 4 2 1 y 5 y y2 1 y 1 5 y y 1 x 5 5 ? 72 2 2 9 3 9 3 2. ⇒ m.m.c. (x, y) 5 2 ? 5 ? 7 5 2 450 y 5 2 ? 52 ? 7 l) x3y 2 y 5 y (x3 2 1) 5 y (x 2 1) (x2 1 x 1 1)
j) y3 1
m) ax2 2 a 1 bx2 2 b 5 5 a (x2 2 1) 1 b (x2 2 1) 5 5 (x2 2 1) (a 1 b) 5 5 (x 1 1) (x 2 1) (a 1 b) 2. 5x2 2 10xy 1 5y2 5 5 (x2 2 2xy 1 y2) 5 5 5 (x 2 y)2 v.n. 5 5 ? 62 5 5 ? 36 5 180 3. ab2 2 ac2 1 b3 2 bc2 5 5 a (b2 2 c2) 1 b (b2 2 c2) 5 5 (b2 2 c2) (a 1 b) 5 (b 1 c) (b 2 c) (a 1 b) 4. x3y 1 2x2y2 1 xy3 5 xy (x2 1 2xy 1 y2) 5 5 xy (x 1 y)2 v.n. 5 10 ? (25)2 5 10 ? 25 5 250 5. ax3 2 ax 1 bx3 2 bx 5 5 ax (x2 2 1) 1 bx (x2 2 1) 5 5 (x2 2 1) (ax 1 bx) 5 5 (x 1 1) (x 2 1) x (a 1 b) 5 5 x (a 1 b) (x 1 1) (x 2 1)
3.
4.
x 5 23 ? 3 ? 11 3 2 ⇒ m.m.c. (x, y) 5 2 ? 3 ? 5 ? 11 5 3 960 y 5 32 ? 5 ? 11
a 5 25 ? 53 7 3 ⇒ m.m.c. (a, b) 5 2 ? 5 5 16000 b 5 27 ? 52
5. a) b) c) d) e) f) g) h) i) j) l)
m.m.c. (xy6, x4y5) 5 x4y6 m.m.c. (a5x2, a2y) 5 a5x2y m.m.c. (xy3, x2y2, x4y) 5 x4y3 m.m.c. (3x6, 5x4) 5 15x6 m.m.c. (ab3, a4c2, bc) 5 a4b3c2 m.m.c. (x3y3, x4y2, x2y5) 5 x4y5 m.m.c. (9x3, 6ax2) 5 18ax3 m.m.c. (4a, 6a2b, 9b3) 5 36a2b3 m.m.c. (18a2b3, 24ab4) 5 72a2b4 m.m.c. (12b2c, 16bc5, 20b3c) 5 240b3c5 m.m.c. (15m3x, 10mx3, 20m2x2) 5 5 60m3x3 m) m.m.c. (14a2p6, 21a4p3, 42a5p5) 5 42a5p6
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