Operações com polinômios resolução

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1. Operações com Polinômios – 8º ano 1) Efetue as operações indicadas: a) (2x3 – 3x2 + x – 1) + (5x3 + 6x2 – 7x + 3) 3 2 3 2 3 3 2 2 3 2 2 3 1 5 6 7 3 2 5 3…

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  • 1. Operações com Polinômios – 8º ano 1) Efetue as operações indicadas: a) (2x3 – 3x2 + x – 1) + (5x3 + 6x2 – 7x + 3) 3 2 3 2 3 3 2 2 3 2 2 3 1 5 6 7 3 2 5 3 6 7 1 3 7 3 6 2 x x x x x x x x x x x x x x x                  b) (– 8y2 – 12y + 5) + (7y2 – 8) gabarito incorreto 2 2 2 2 2 8 12 5 7 8 8 7 12 5 8 8 12 3 y y y y y y y y              c) (2ax3 – 5a2 x – 4by) + (5ax3 + 7a2 x + 6by) 3 2 3 2 3 3 2 2 3 2 2 5 4 5 7 6 2 5 5 7 4 6 7 2 2 ax a x by ax a x by ax ax a x a x by by ax a x by             d) (a2 – b2 ) + (a2 – 3b2 – c) + (5c – 2b2 – a2 ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 5 2 3 2 5 6 4 a b a b c c b a a a a b b b c c a b c                 e) (3y2 – 2y – 6) – (7y2 + 8y + 5) 2 2 2 2 2 3 – 2 – 6 7 8 5 3 7 – 2 8 – 6 5 4 10 11 y y y y y y y y y y          f) (8x3 – 4x2 + 3x – 5) – (6x3 – 7x2 + 5x – 9) 3 2 3 2 3 3 2 2 3 2 8 4 3 5 6 7 5 9 8 6 4 7 3 5 5 9 2 3 2 4 x x x x x x x x x x x x x x x                  g) (2x3 – 3x + 1) – (– 4x2 + 3) 3 2 3 2 2 3 1 4 3 2 4 3 2 x x x x x x        h) (2x3 – 5x2 + 8x – 1) – (– 3x3 + 5x2 – 5x + 6) 3 2 3 2 3 3 2 2 3 2 2 5 8 1 3 5 5 6 2 3 5 5 8 5 1 6 5 10 13 7 x x x x x x x x x x x x x x x                  i) (x2 – 5xy + y2 ) + (3x2 – 7xy + 3y2 ) – (4y2 – x2 ) 2 2 2 2 2 2 2 2 2 2 2 2 2 5 3 7 3 4 3 5 7 3 4 5 12 x xy y x xy y y x x x x xy xy y y y x xy                j)    2 2 2 2 21 4 5 3 5 2 ab a ab ab a            2 2 2 2 2 2 2 2 2 2 2 1 4 5 3 5 2 2 2 3 9 2 9 2 2 2 ab ab ab a a ab ab ab ab a a              k)              1 3 2 2 1 1 2 1 3 2 22 mmmm 2 2 2 2 2 2 2 2 1 1 2 1 1 3 2 2 3 2 1 1 2 1 1 3 2 2 3 4 3 3 4 6 6 m m m m m m m m m m m m m m                l)              2 12 3 1 2 1 m mnmnm 1 1 2 1 2 3 2 1 1 2 1 2 2 3 3 2 3 3 3 3 m m mn mn m m mn mn mn mn mn                m)              baabbaab 2222 4 3 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 3 2 3 4 3 1 2 4 3 8 3 3 5 2 4 3 4 3 ab a b ab a b ab ab a b a b ab ab a b a b ab a b           
  • 2. 2) Efetue as multiplicações: a) y(4x2 – 2x3 – 7) 2 3 4 2 7x y x y y  b) (x4 – 3x2 – 5x + 1)(– 4x) 5 3 2 4 12 20 4x x x x    c) 2x(y2 + xy + 1) 2 2 2 2 2xy x y x  d) 4ab(a2 + b2 – ab) 3 3 2 2 4 4 4a b ab a b  e) 4xy2 (4x + y + 1) 2 2 3 2 16 4 4x y xy xy  f)  3 2 1 2 1 2 x x x x            4 3 21 1 1 2 2 2 x x x x   g)  2 5 3 6 5 3 ab ab a          2 2 2 25 5 10 3 a b a b a   h) (2x + 3)(5x – 1) 2 2 10 2 15 3 10 13 3 x x x x x      i) (4x3 + 2x – 3)(5x2 + x – 1) 5 4 3 3 2 2 5 4 3 2 20 4 4 10 2 2 15 3 3 20 4 6 13 5 3 x x x x x x x x x x x x x              j) (x2 – 2x + 5)(x3 – 3x2 + 6) 5 4 2 4 3 3 2 5 4 3 2 3 6 2 6 12 5 15 30 5 9 30 x x x x x x x x x x x x             3) Calcule os seguintes quocientes: a) (6ax – 9bx – 15x) : 3x 6 9 15 2 3 5 3 3 3 ax bx x a b x x x      b) (8a2 – 4ac + 12a) : 4a 2 8 4 12 2 3 4 4 4 a ac a a c a a a      c) (27ab – 36bx – 36by) : (– 9b) 27 36 36 3 4 4 9 9 9 ab bx by a x y b b b          d) (49an – 21n2 – 91np) : 7n 2 49 21 91 7 3 13 7 7 7 an n np a n p n n n      e) (27a2 bc – 18acx2 – 15ab2 c) : (– 3ac) 2 2 2 2 227 18 15 9 6 5 3 3 3 a bc acx ab c ab x b ac ac ac          f) (8x5 y + 4x3 y2 – 6x2 y): (4x2 y) 5 3 2 2 3 2 2 2 8 4 6 3 2 4 4 4 2 x y x y x y x xy x y x y x y      g)          3 4 :20812 2 a axyabxxa 2 2 12 3 8 3 20 3 1 4 1 4 1 4 36 24 60 9 6 15 4 4 4 a x abx axy a a a a x abx axy ax bx xy a a a           h) 6 : 4 1 3 1 2 1 ab abcabyabx        6 6 6 2 3 4 6 6 6 2 3 2 2 3 4 3 2 abx aby abc ab ab ab abx aby abc x c y ab ab ab          
  • 3. 4) Determine o quociente e o resto das seguintes divisões: a) (4a2 – 7a + 3) : (4a – 3) 2 2 4 7 3 4 3 4 3 1 4 3 4 3 0 a a a a a a a a          b) (11x2 – 2 – x + 10x3 ) : (5x – 2) 3 2 3 2 2 2 2 10 11 2 5 2 10 4 2 3 1 15 15 6 5 2 5 2 0 x x x x x x x x x x x x x x               c) (7x – 2x4 + 3x5 – 2 – 6x2 ) : (3x – 2) 5 4 2 5 4 4 2 2 3 2 6 7 2 3 2 3 2 2 1 6 7 6 4 3 2 3 2 x x x x x x x x x x x x x x x                0 d) (x3 – 2x2 – 6x – 27) : (x2 – 5x + 9) 3 2 2 3 2 2 2 2 6 27 5 9 5 9 3 3 15 27 3 15 27 54 x x x x x x x x x x x x x                e) (x2 + 5x + 10) : (x + 2) 2 2 5 10 2 2 3 3 10 3 6 4 x x x x x x x x          f) (10x – 9x2 + 2x3 – 2) : (x2 + 1 – 3x) 3 2 2 3 2 2 2 2 9 10 2 3 1 2 6 2 2 3 3 8 2 3 9 3 1 x x x x x x x x x x x x x x                  g) (6x3 – 16x2 + 5x – 5) : (2x2 + 1 – 4x) 3 2 2 3 2 2 2 6 16 5 5 2 4 1 6 12 3 3 2 4 2 5 4 8 2 6 3 x x x x x x x x x x x x x x                 h) (x6 + 4x3 + 2x – 8) : (x4 + 2x2 + 4) 6 3 4 2 6 4 2 2 4 3 2 4 2 3 4 2 8 2 4 2 4 2 2 + 4 4 2 8 2 4 8 4 2 x x x x x x x x x x x x x x x x x                 
  • 4. 03/08/2016 23:24:05-D:EscolaMarista20168º ano - MatemáticaII TrimestreAV1Operações com Polinômios - Resolução.docx 4 2 2 ) 3 2 4 5 3 2 5 3 9 2 a x x x x x x                   2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 ) 3 2 4 5 3 5 3 2 5 15 9 10 6 20 12 10 25 6 15 15 9 10 6 20 12 10 25 6 15 15 9 10 10 6 20 25 6 12 15 15 9 45 27 b x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x                                       5) Qual o polinômio que, ao ser dividido por x – 6, tem quociente 2x – 5 e resto – 12?     2 2 6 2 5 12 2 5 12 30 12 2 17 18 x x x x x x x           6) Determine o polinômio que, dividido por x – 1, tem quociente x – 1 e resto 2.     2 2 1 1 2 2 1 2 2 3 x x x x x x           7) O quociente da divisão de um polinômio A por x2 – 2x + 1 é x2 + 4x + 3. O resto dessa divisão é 12x + 3. Qual é o polinômio A?    2 2 4 3 2 3 2 2 4 3 3 2 2 2 4 3 2 2 1 4 3 12 3 4 3 2 8 6 4 3 12 3 4 2 3 8 6 4 12 3 3 2 4 10 6 x x x x x x x x x x x x x x x x x x x x x x x x x x x                                8) A divisão de dois polinômios é exata. O quociente dessa divisão é x2 – 7x + 12 e o polinômio divisor é x2 – 5. Qual é o polinômio dividendo?    2 2 4 2 3 2 4 3 2 2 4 3 2 7 12 5 5 7 35 12 60 7 5 12 35 60 7 7 35 60 x x x x x x x x x x x x x x x x x                   9) Dados os polinômios A = 3x2 + 2x – 4, B = 5x – 3 e C = 2x + 5, calcule: a) A + B + C b) AB – BC c) A2 – 2B      2 2 4 3 2 3 2 2 4 3 3 2 2 2 4 3 2 ) 3 2 4 3 2 4 2 5 3 9 6 12 6 4 8 12 8 16 10 6 9 6 6 12 4 12 8 8 10 16 6 9 12 20 26 22 c x x x x x x x x x x x x x x x x x x x x x x x x x x x                                RESPOSTAS: 1) a) 7x3 +3x2 -6x+2 b) -y2 -12y-3 c) 7ax3 +2a2 +2by d) a2 -6b2 +4c e) -4y2 -10y-11 f) 2x3 +3x2 -2x+4 g) 2x3 +4x2 -3x-2 h) 5x3 -10x2 +13x-7 i) 5x2 -12xy j)3ab2 /2+9a2 -2 k) m2 /6+m/6 l) 2mn/3+3 m)5ab2 /4+2a2 b/3 2) a) -6x3 y+12x2 y-21y b) -4x5 +12x3 +20x2 -4x c) 2xy2 +2x2 y+2x d) 4a3 b+4ab3 -4a2 b2 e) 16x2 y2 +4xy+4xy2 f) x4 /2+x3 /2-x2 -x/2 g) -5a2 b+10a2 b2 -25/3a h) 10x2 +13x-3 i) 20x5 +4x4 +6x3 -13x2 -5x+3 j) x5 -5x4 +11x3 -9x2 -12x+30 3) a) 2a-3b-5 b) 2a-c+3 c)-3a+4x+4y d) 7a-3n-13p e)-9ab+6x2 +5b2 f) 2x3 +xy-3/2 g) 9ax-6bx+15xy h) 3x-2y+3/2 c 4) a) Q=a-1 e R=0 b) Q=2x2 +3x+1 e R=0 c) Q=x4 -2x+1 e R=0 d) Q=x+3 e R=-54 e) Q=x+3 e R=4 f) Q=2x-3 e R=-x+1 g) Q=3x-2 e R=-6x-3 h) Q=x2 -2 e R=4x3 +2x 5) 2x2 -17x+18 6) x2 -2x+3 7) x4 +2x3 -4x2 +10x+6 8) x4 -7x3 +7x2 +35x-60 9) a) 3x2 +9x-2 b) 15x3 -9x2 -45x+27 c)9x4 +12x3 -20x2 -26x+22
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