Factor Comun

1472. Scoateţi factor comun: a = 2 ⋅ 5 + 5 ⋅ 7;

b = 2 ⋅ 5 + 2 2 ⋅ 7;

c = 2 ⋅ 5 + 2 2 ⋅ 7 − 2 ⋅ 3;

d = 22 ⋅ 3 + 27 ;

e = 5 3 + 5 2 − 5;

f = 10 2 ⋅ 51 −10 2 ⋅ 49;

g = 2x + 2y;;

h = 2x + 4y − 6z;

i = 10 2 ⋅ 5 −10 2 ⋅ 3 + 2a + 2b.

1473. Să se verifice că: a ) 2 + 2 8 + 2 7 + 2 6 = 30 ⋅ 2 5 ; 9

b) 214 ⋅ (315 + 316 + 317 + 318 ) = 314 ⋅ (217 + 218 + 219 + 2 20 ); c ) 2(a + 2) + b(2 + a) = a(2 + b) + 2(b + 2); 2

15

3

d) 2 n ⋅ 3 n+1 + 2n +1 ⋅ 3 n+2 − 2 n+2 ⋅ 3 n = [2 8 ⋅ 3 4 : (2 21 ⋅ 3 21 )3 ]n ⋅ (2 ⋅ 3 2 − 1); e ) 9 8 − 8 ⋅ 9 7 − 8 ⋅ 9 6 − 8 ⋅ 9 5 − 8 ⋅ 9 4 − 8 ⋅ 9 3 − 8 ⋅ 9 2 − 8 ⋅ 9 − 1 = 8 ⋅ (G.M. 4/1978) f ) 2m+1 ⋅ 3 m ⋅ 5 m +2 − 2m ⋅ 3 m +2 ⋅ 5 m+1 = 5 ⋅ 30 m ; m ∈ N; g) 2116 − 2115 + 2114 − 2113 + ... + 2102 − 2101 = 5 ⋅ 17 ⋅ 257 ⋅ 2101 ; h) (a ⋅ b n+1 )m ⋅ (b ⋅ a m+1 )n + (ab) m(n +1) : (ab)n = (ab) m(n +1) ⋅ [(ab)n + 1 : (ab) n ].

192. Folosind şi metoda factorului comun simplificaţi fracţiile următoare: f1 =

9a + 9b ; 24a + 24b

f2 =

56 ⋅ 35 + 247 ⋅ 56 ; 174 ⋅ 94 + 294 ⋅ 94

3 5 7 + + 2m + 3n − 5 10 14 18 f4 = ; f5 = ; 9 15 21 10m + 15n − 25 + + 20 28 36

f6 =

f3 =

24 ⋅ 7 + 21⋅ 4 + 28 ⋅ 3 ; 36 ⋅ 5 + 45 ⋅ 2 + 18 ⋅ 25

5(m + 1) + 7(m + 1) + b(m + 1) . 17(m + 1) + 7(m + 1) + 2b(m + 1)

141. Folosind şi metoda factorului comun să se simplifice fracţiile: 42 ⋅ 31 + 42 ⋅ 29 ; 105 ⋅ 67 + 105 ⋅ 13

f1 =

6a + 6b ; 12a + 24

f4 =

8a + 12 ; 12a + 18

f7 =

123 + 246 + 369 ; 1230

f8 =

123123123 ; 246246246

f11 =

f10 =

f2 =

f5 =

2a + 4b + 6c ; 4a + 8b + 12c

12 + 24 + 48 + 12 2 ; 31 + 62 + 124 + 312 200802008 ; 1606416064

2x + 4y

f3 = 5x + 10y ; f6 =

f9 =

1234 ; 2468

103103 ; 1031030

f12 =

1223136693 . 2121363639

Exemplu nr. 3: Scoateţi factor comun: a = + 27 – 36 + 81 – 72; b = – 24 + 48 – 96 + 72; c =+125 – 75 +50 – 25; d = – 88 + 66 – 22 + 110 ; e = – 125 + 75 – 150 – 225 + 600 – 325 + 1775. Exemplu nr. 5; Scoateţi factor comun: a = 36 – 3x; b = 45x – 15; c = 8x – 4x – 16; d = 3x – 2x; e = 5x – 7x + + 11x; f = 9x – 3x; g = 14 x – 7x; h = 9x – 27x – 36x. Scoateţi factor comun şi efectuaţi în paranteză (reduceţi termenii asemenea):

s1 = – 35 +15 – 40 + 55 – 60 = (+5)·( – 7 + 3 – 8 +11–12) = (+ 5)·(–13) = – 65. s2 = + 24 – 60 + 48 – 72 = (+12) . ( 2 – 5 + 4 – 6 ) = (+ 12) . (– 5) = – 60. s3 = +3x – 2x = x·(+3 – 2) = x·(+ 1) = + 1·x = + x. s4 = + 7y – 7y =(+ y)·(7 –7 )= 0. s5 = +9x – 18x + 27x = + 9x·(+ 2) = + 18x. s5 = + 3z – 2z + 4z – 6z = (+z)·(+3 –2 + 4 – 6) = (+z)·( –1) = – z. s6 = +9x – 18x + 27x – 36x + 45x – 54x = +9x·(1 – 2 + 3 – 4 + 5 – 6) = – 27x. s7 = (2mx – m) + (2px – p) = m(2x – 1) + p(2x – 1) = (2x – 1) (m + p). Cercetaţi dacă putem scoate factor comun: a) pe +6 din s1 = +18 + 30 – 72 + 726; pe –15 din s2 = – 45 + 90 – 195 + 240; 465. Pentru suma: s = + 3360 – 5040 + 11760 – 18480 scoateţi factor comun pe: a) – 4; b) + 5; c) – 7; d) + 48; e) cel mai mare factor comun. 468. Scoateţi factor comun: a = 75 – 5x; b = 27x – 81; c = 24x – 36x + 72; d = 5x – 4x; e = 7x – 9x + 13x; f = 28x – 7x; g = 32x – 24x; h = 42x – 56x + 63x; 2 I = – 9x + 7x – 6x + 15x; j = +8x – 4x2 + 12x2 – 16x2; k = 5ab + 3a2b – 2ab2. 469. Scoateţi factor comun: a = 2xy – 4x; b = 2xy – 4y; c = 2xy – 4yz; d = 8xyz – 8xyz; e = 10xz – 15yz; f = 4x2 – 4x; g = 19x2 – 38x3; h = 30x2y – 15x2; i = 60xy2 – 45y2; j = 48xz2 – 60x2y; k = 50xy2z2 – 75x2yz2 + 125x2y2z; l = (3x2y)3 –(9xy2)2; m = 4x2y3z4 – 16x3y4z5 + 12x4y5z6;

n = 2·(a +1) – 3·(a + 1) + 7·(a + 1).

470. Scoateţi factor comun şi efectuaţi in paranteză (reduceţi termenii asemenea): a = + 9 – 12 + 15 – 18; b = +25 – 75 + 100 – 125; c = – 64 + 80 – 88 + 72; d = 4x – 5x + 7x; e = 2x – 4x + 2x; f = 5x – 4x + 6x – 7x + 19x – 19x ; g = – 7z + 8z + 11z–22z+31z–36z; h = 4y – 6y + 8y – 10y + 12y – 14y. 471. Scoateţi factor comun: a = 22·3 – 2·32; b = 22·34 – 24·33 + 23·34 – 24·32; c = 2·32··5 + 22·3·52 – 22·32·5 + 22·33·52; d = 5m·7m+1 – 5m+1·7n; e = 2m·3 – 2m·5; f = 2m·5 + 2m·7 – 2m·11; g = 2m·3 + 2m+1 – 2m+2 – 2m; h = 2m+1·5 + 2m+2·7 – 2m+3·3; i = 2m·3n – 2m+1·3n + 2m·3n+1; j = 2m·3n·5 – 2m·3n·7 + 2m+1·3n – 2m·3n+1; k = 2m·3n – 22·33 +2m+1·3n+2 .