Ejercicio 47

Ejercicio 47

• Se multiplica el primer polinomio por el segundo, en cada grupo
• Se suma ambos resultados de la multiplicacion
• Reduciomos términos semejantes

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3. a( a–x ) +3a( x+2a ) –a( x–3a ) a( a–x ) +3a( x+2a ) –a( x–3a ) = a 2 –ax+3ax+6 a 2 –ax+3 a 2 = 10 a 2 +ax
4. x 2 ( y 2 +1 ) + y 2 ( x 2 +1 ) –3 x 2 y 2 x 2 ( y 2 +1 ) + y 2 ( x 2 +1 ) –3 x 2 y 2 = x 2 y 2 + x 2 + x 2 y 2 + y 2 –3 x 2 y 2 = x 2 – x 2 y 2 + y 2
5. 4 m 3 –5m n 2 +3 m 2 ( m 2 + n 2 ) –3m( m 2 – n 2 ) 4 m 3 –5m n 2 +3 m 2 ( m 2 + n 2 ) –3m( m 2 – n 2 ) = 4 m 3 –5m n 2 +3 m 4 +3 m 2 n 2 –3 m 3 +3m n 2 = 3 m 4 + m 3 +3 m 2 n 2 –2m n 2
6. y 2 + x 2 y 3 – y 3 ( x 2 +1 ) + y 2 ( x 2 +1 ) – y 2 ( x 2 –1 ) y 2 + x 2 y 3 – y 3 ( x 2 +1 ) + y 2 ( x 2 +1 ) – y 2 ( x 2 –1 ) = y 2 + x 2 y 3 – x 2 y 3 – y 3 + y 2 x 2 + y 2 – y 2 x 2 + y 2 = – y 3 +3 y 2
7. 5( x+2 ) –( x+1 ) ( x+4 ) –6x 5( x+2 ) –( x+1 ) ( x+4 ) –6x = 5x+10–( x 2 +4x+x+4 ) –6x = 5x +10– x 2 – 5x –4–6x = 6–6x– x 2
8. ( a+5 ) ( a–5 ) –3( a+2 ) ( a–2 ) +5( a+4 ) ( a+5 ) ( a–5 ) –3( a+2 ) ( a–2 ) +5( a+4 ) = a 2 – 5a + 5a –25–3( a 2 – 2a + 2a –4 ) +5a+20 = a 2 –5–3 a 2 +12+5a = 7+5a–2 a 2
9. ( a+b ) ( 4a–3b ) –( 5a–2b ) ( 3a+b ) –( a+b ) ( 3a–6b ) ( a+b ) ( 4a–3b ) –( 5a–2b ) ( 3a+b ) –( a+b ) ( 3a–6b ) = 4 a 2 –3ab+4ab–3 b 2 –( 15 a 2 +5ab–6ab–2 b 2 ) –( 3 a 2 –6ab+3ab–6 b 2 ) = 4 a 2 +ab–3 b 2 –( 15 a 2 –ab–2 b 2 ) –( 3 a 2 –3ab–6 b 2 ) = 4 a 2 +ab–3 b 2 –15 a 2 +ab+2 b 2 –3 a 2 +3ab+6 b 2 = –14 a 2 +5ab+5 b 2
10. ( a+c ) 2 – ( a–c ) 2 ( a+c ) 2 – ( a–c ) 2 = ( a+c ) ( a+c ) –( a–c ) ( a–c ) = a 2 +ac+ac+ c 2 –( a 2 –ac–ac+ c 2 ) = a 2 +2ac+ c 2 – a 2 +2ac– c 2 = 4ac
11. 3 ( x+y ) 2 –4 ( x–y ) 2 +3 x 2 –3 y 2 3 ( x+y ) 2 –4 ( x–y ) 2 +3 x 2 –3 y 2 = 3( x 2 +2xy+ y 2 ) –4( x 2 –2xy+ y 2 ) +3 x 2 –3 y 2 = 3 x 2 +6xy+ 3 y 2 –4 x 2 +8xy–4 y 2 +3 x 2 – 3 y 2 = 2 x 2 +14xy–4 y 2
12. ( m+n ) 2 – ( 2m+n ) 2 + ( m–4n ) 2 ( m+n ) 2 – ( 2m+n ) 2 + ( m–4n ) 2 = ( m+n ) ( m+n ) –( 2m+n ) ( 2m+n ) +( m–4n ) ( m–4n ) = m 2 +mn+mn+ n 2 –( 4 m 2 +2mn+2mn+ n 2 ) +( m 2 –4mn–4mn+16 n 2 ) = m 2 +2mn+ n 2 –4 m 2 –4mn– n 2 + m 2 –8mn+16 n 2 = –2 m 2 –10mn+16 n 2
13. x( a+x ) +3x( a+1 ) –( x+1 ) ( a+2x ) – ( a–x ) 2 x( a+x ) +3x( a+1 ) –( x+1 ) ( a+2x ) – ( a–x ) 2 = ax+ x 2 +3ax+3x–( ax+2 x 2 +a+2x ) –( a 2 –2ax+ x 2 ) = ax + x 2 +3ax+3x– ax –2 x 2 –a–2x– a 2 +2ax– x 2 = –2 x 2 +5ax+x–a– a 2
14. ( a+b–c ) 2 + ( a–b+c ) 2 – ( a+b+c ) 2 ( a+b–c ) 2 + ( a–b+c ) 2 – ( a+b+c ) 2 = ( a+b–c ) ( a+b–c ) +( a–b+c ) ( a–b+c ) –( a+b+c ) ( a+b+c ) = a 2 + ab – ac + ab + b 2 –bc– ac –bc+ c 2 + a 2 – ab + ac – ab + b 2 –bc+ ac –bc+ c 2 –( a 2 +ab+ac+ab+ b 2 +bc+ac+bc+ b 2 ) = 2 a 2 +2 b 2 –4bc+2 c 2 –( a 2 +2ab+2ac+2bc+ b 2 + c 2 ) = 2 a 2 +2 b 2 –4bc+2 c 2 – a 2 –2ab–2ac–2bc– b 2 – c 2 = a 2 –2ab–2ac+ b 2 –6bc+ c 2
15. ( x 2 +x–3 ) 2 – ( x 2 –2+x ) 2 + ( x 2 –x–3 ) 2 ( x 2 +x–3 ) 2 – ( x 2 –2+x ) 2 + ( x 2 –x–3 ) 2 = ( x 2 +x–3 ) ( x 2 +x–3 ) –( x 2 –2+x ) ( x 2 –2+x ) +( x 2 –x–3 ) ( x 2 –x–3 ) = x 4 + x 3 –3 x 2 + x 3 + x 2 –3x–3 x 2 –3x+9–( x 4 –2 x 2 + x 3 –2 x 2 +4–2x+ x 3 –2x+ x 2 ) + x 4 – x 3 –3 x 2 – x 3 + x 2 +3x–3 x 2 +3x+9 = x 4 +2 x 3 –5 x 2 –6x+9–( x 4 –3 x 2 +2 x 3 +4–4x ) + x 4 –2 x 3 –5 x 2 +6x+9 = x 4 + 2 x 3 –5 x 2 – 6x +9– x 4 +3 x 2 – 2 x 3 –4+4x+ x 4 –2 x 3 –5 x 2 + 6x +9 = x 4 –2 x 3 –7 x 2 +4x+14
16. ( x+y+z ) 2 –( x+y ) ( x–y ) +3( x 2 +xy+ y 2 ) ( x+y+z ) 2 –( x+y ) ( x–y ) +3( x 2 +xy+ y 2 ) = ( x+y+z ) ( x+y+z ) –( x 2 – xy + xy – y 2 ) +3 x 2 +3xy+3 y 2 = x 2 +xy+xz+xy+ y 2 +yz+xz+yz+ z 2 – x 2 + y 2 +3 x 2 +3xy+3 y 2 = 3 x 2 +5 y 2 + z 2 +5xy+2xz+2yz
17. [ x+( 2x–3 ) ] [ 3x–( x+1 ) ] +4x– x 2 [ x+( 2x–3 ) ] [ 3x–( x+1 ) ] +4x– x 2 = [ x+2x–3 ] [ 3x–x–1 ] +4x– x 2 = [ 3x–3 ] [ 2x–1 ] +4x– x 2 = 6 x 2 –3x–6x+3+4x– x 2 = 5 x 2 –5x+3
18. [ 3( x+2 ) –4( x+1 ) ] [ 3( x+4 ) –2( x+2 ) ] [ 3( x+2 ) –4( x+1 ) ] [ 3( x+4 ) –2( x+2 ) ] = [ 3x+6–4x–4 ] [ 3x+12–2x–4 ] = [ –x+2 ] [ x+8 ] = – x 2 –8x+2x+16 = – x 2 –6x+16
19. [ ( m+n ) ( m–n ) –( m+n ) ( m+n ) ] [ 2( m+n ) –3( m–n ) ] [ ( m+n ) ( m–n ) –( m+n ) ( m+n ) ] [ 2( m+n ) –3( m–n ) ] = [ m 2 – mn + mn – n 2 –( m 2 +mn+mn+ n 2 ) ] [ 2m+2n–3m+3n ] = [ m 2 – n 2 – m 2 –2mn– n 2 ] [ –m+5n ] = [ –2mn–2 n 2 ] [ –m+5n ] = 2 m 2 n–10m n 2 +2m n 2 –10 n 3 = 2 m 2 n–8m n 2 –10 n 3
20. [ ( x+y ) 2 –3 ( x–y ) 2 ] [ ( x+y ) ( x–y ) +x( y–x ) ] [ ( x+y ) 2 –3 ( x–y ) 2 ] [ ( x+y ) ( x–y ) +x( y–x ) ] = [ ( x+y ) ( x+y ) –3( x–y ) ( x–y ) ] [ x 2 – xy + xy – y 2 +xy– x 2 ] = [ x 2 +xy+xy+ y 2 –3( x 2 –xy–xy+ y 2 ) ] [ – y 2 +xy ] = [ x 2 +2xy+ y 2 –3 x 2 +6xy–3 y 2 ] [ – y 2 +xy ] = [ –2 x 2 +8xy–2 y 2 ] [ – y 2 +xy ] = 2 x 2 y 2 –2 x 3 y–8x y 3 +8 x 2 y 2 +2 y 4 –2x y 3 = –2 x 3 y+10 x 2 y 2 –10x y 3 +2 y 4