Chuyên ĐS-8 HKI Quang Duy (093.50.30.798) § NHÂN ĐƠN THỨC VỚI ĐA THỨC Quy tắc: A(B + C) = AB + AC 1) 8x.( 3x3 – 6x +4 ) = 8x.3x3 +8x.( –6x) +8x.4= 24 x4 – 48x2 + 32x 1 2) 2x2.(x2 + 5x – ) = 2x3.x2 + 2x3.5x – 2x3 = 2x5 + 10x4 – x3 2 3) ( 3x3y – x + xy ).6 xy = 18x4 y4 – 3x3y3 + x2y4 5 4) (4x3 – 5xy + 2x) (– xy) = –2x4 y + x2y2 – x2y 2 Áp dụng: xn-1(x + y) –y(xn-1yn-1) = xn-1.x + xn-1.y – xn-1.y – y.yn-1 = xn-1+1 + xn-1.y – xn-1.y – y1+n+1 VD: § NHÂN ĐA THỨC VỚI ĐA THỨC Quy tắc: (A + B)(C + D) = AC + AD + BC + BD VD: Tính 1) (x + 3)(x2 + 3x –5) 2) (xy–1) ( xy+5) 3) (2x –5)(3x2 + 7x –1) xy –1)(x3 –2x –6) Áp dụng: (x – y) (x2 + xy + y2) 4) ( = x3 +3x2 –5x +3x2 + 9x–15 = x3 + 6x2 +4x –15 = x2y2 + 5xy – xy –5 = x2y2 + 4xy – = 2x(3x2 + 7x – 1) – 5( 3x2 + 7x – 1) = 6x3 +14x2 – 2x – 15x2 – 35x+5 = 6x3 – x2 – 37x + = x4 y –x2y –3xy –x3 +2x + = x (x2 + xy + y2) – y (x2 + xy + y2) = x3 + x2y + xy2 – x2y – xy2 – y3 = x3 – y3 § NHỮNG HẰNG ĐẲNG THỨC ĐÁNG NHỚ 1) (A + B)2 = A2 + 2AB + B2 2) (A – B)2 = A2 – 2AB + B2 3) A2 – B2 = (A + B)(A – B) 4) (A + B)3= A3+ 3A2B + 3AB2 + B3 5) (A – B)3= A3– 3A2B + 3AB2 – B3 6) A3 + B3 = (A + B)(A2– AB + B2) 7) A3 – B3 = (A – B)(A2 + AB + B2) Chuyên ĐS-8 HKI Quang Duy (093.50.30.798) Bài Nhân đơn thức với đa thức: 1) 3x2(5x2 – 2x – 4) 2) xy2(x2y + x3y2 + 3x2y3) 4) 2x2(4x2 − 5xy + 8y3) 7) (3xy – x2 + y) 2 xy 10) – x4y2(6x4 − 10 xy – y5) x – 4x) Bài Nhân đa thức với đa thức: 1) (2x − 5)(3x + 7) 4).(x + 3)(2x2 + x − 2) 7).(3x + 4x2 − 2)(− x2 +1 + 2x) 10).(x – 2)(3x2 – 2x + 1) 13).(xy – 1)(x2y – 3xy2) 16).(x2 – 2xy – y2)(x – y) 19) (2x2 – 1)(3x2 – x + 2) 22) (7x – 1)(2x2 – 5x + 3) 13) 3x(2x3 – 25) (− x2+y3)(8x3 − xy –y2) 3) xyz(x2y + 3yz2 + 4xy2z) 5) 2xy2(5x2 + 3xy − 6y3) 6) – x2y(xy2 – xy + x2y2) 1 8) (4x3 – 5xy + 2x)( – xy) 9) 2x2(x2 + 3x + ) 2 11) x3(x + x2 – x5) 12) 2xy2(xy + 3x2y – xy3) 3 10 4 14) x3y5(7x4 + 5x2y − x y –y ) 21 2) (−3x + 2)(4x − 5) 5) (2x − y)(4x2 − 2xy + y2) 8) (2x – y)(4x2 + 2xy + y2) 11).(x + 2)(x2 + 3x + 2) 14) (x + 3)(x2 – x + 2) 17) (x2 – 3xy + y2)(x + y) 20) (2 – 3x2)(x3 + 2x2 – 3) 23) (5x + 3)(3x2 + 6x + 7) 26) (2xy2−7x2y)( x + 5xy 3) (x − 2)(x2 + 3x − 1) 6) (x +3)(x2 –3x + 9) – (54 + x3) 9) (2x + y)(4x2 – 2xy + y2) 12.) (2x2 + 1)(x2 – x +3) 15) (x2 – x + 2)(2x – 3) 18) (x – 5)(x2 – 6x + 1) 21) (9x – 2)(x2 – 3x + 5) 24) (6x2 + 5y2)(2x2– y2) − 4y3) Bài Rút gọn tính giá trị biểu thức: 1) A = 5x(4x2 – 2x + 1) – 2x(10x2 – 5x – 2) với x= 15 2) 2x (3x2 − 5x + 8) − 3x2(2x − ) – 16x với x = − 15 2 3) B = 5x(x – 3) + x (7 – 5x) – 7x với x = – 4) C = (x – 2)(x4 + 2x3 + 4x2 + 8x +16) với x = 5) D = 4x – 28x + 49 với x = 6) E = x3 – 15x2 + 75x với x = 25 7) F = (x + 1)(x – 1)( x2 + x + 1)( x2 – x + 1) với x = 8) G = x(x – y) + (x + y) với x = y =8 9) H = 5x(x – 4y) – 4y(y – 5x) với x= – 1/5; y= –1/2 2 2 10) I = x(x – y ) – x (x + y) + y(x – x) với x = 1/2 y = 100 11) J = (x + y)(x3 – x2y + xy2 – y3) với x = y = – 1/2 2 12) K = 4x (5x – 3y) – 5x (4x + y) với x = –2; y = –3 13) L = (x2y + y3)(x2 + y2) – y(x4+ y4) với x = 0,5; y = – 2 2 14) (2x + y) (x − 6xy ) − 2x (x – 3y ) (x + ) + 6x y (y − 2x) với x = − | y| = Bài Hằng đẳng thức: 1) (x – 2y)2 2) (x + 2y)2 3) (3x + y)2 4) (3x – y)2 2 5) (2x + 5y) 6) (2x – 5y) 7) (2x – 3y) 8) (2x + 3y)2 9) (x – 3)2 10) (x + 3)2 11) (2x – 1)2 12) (2x +1)2 2 13) (3x – 2) 14) (3x + 2) 15) (4x – 1) 16) (4x + 1)2 2 17) (6x – 3y) 18) (6x + 3y) 19) (3 – 2x) 20) (3 + 2x)2 21) (2x – 3y)3 22) (2x + 3y)3 23) (2x – 1)3 24) (2x + 1)3 2 2 25) (3 + xy ) 26) (2x y – 10) 27) (3x y – 1) 28) (2 – xy2)3 29) x2 – 2x + 30) x2 + 2x + 31) x2 – 4x + 32) x2 + 4x + 2 33) 4x – 4x + 34) 4x + 4x + 35) 9x – 6x + 36) 9x2 + 6x + 37) x2 – 6x + 38) x2 + 6x + 39) x2 – 8x + 16 40) x2 + 8x + 16 2 41) (x – 3) – 16 42.) 64 + 16x + x 43) x – x + 1/4 44) x2 + x + 1/4 45) x2 – 46) 4x2 – 47) 9x2 – 48) 16 – x2 3 49) – x 50) – x 51) 27 – 8x 52) + x3 3 53) 27 + 8x 54) 8x + 64 55) x – 27 56) x3 + 27 57) − 2y + y2 58) 16 – 24y + 9y2 59) − 4x2 60) 100x2 – (x2 + 25)2 61) 27 + 27x + 9x2 + x3 62) – x3 + 3x2 – 3x + Chuyên ĐS-8 HKI Quang Duy (093.50.30.798) 63) x – 3x + 3x – 64) x + 3x + 3x + 65) 8x3 – 12x + 6x – 66) 8x3 + 12x + 6x + 67) – 36x + 54x – 27x 68) + 36x + 54x2 + 27x3 69) 8x3 – 12x2y + 6xy2 – y3 70) 8x3 + 12x2y + 6xy2 + y3 Bài Chứng minh biểu thức sau không phụ thuộc vào x: 1) (3x – 5)(2x + 11) – (2x + 3)(3x + 7) 2) (x – 5)(2x + 3) – 2x(x – 3) + x + 3) (2x + 3)(4x2 – 6x + 9) – 2(4x3 – 1) 4) x(5x – 3) – x2(x – 1) + x(x2 – 6x) – 10 + 3x 5) x(x2 + x + 1) – x2(x +1) – x + 6) x(2x + 1) – x2(x + 2) + x3 – x + Bài Tìm giá trị nhỏ giá trị lớn biểu thức (nếu có): A = x2 – 4x + B = 4x2 + 4x + 11 C = x + 4x + D = – 8x + x2 E = x(x – 6) F = (x – 3)2 + (x – 11)2 G = (x –1)(x + 3)(x + 2)(x + 6) H = (x + 1)(x – 2)(x – 3)(x – 6) I = – 8x – x J = 4x – x2 +1 2 K = x (2– x ) Bài Phân tích đa thức thành nhân tử phương pháp đặt nhân tử chung: 1) 2x2 – 4x 2) 3x – 6y 3) x2 – 3x 4) 4x – 6x 5) x – 4x 6) 9x3y2 + 3x2y2 7) x3 + 2x2 + 3x 8) 6x2y + 4xy2 + 2xy 9) 5x2(x – 2y) – 15x(x – 2y) 10) 3(x – y) – 5x(y – x) 11) 3x(x – 1) + 5(1 – x) 12) 2(2x – 1) + 3(1 – 2x) 13) 10x(x – y) – 8y(y – x) 14) 3x(y + 2) – 3(y + 2) 15) x2 – y2 – 2x + 2y 2 16) 2x + 2y – x – xy 17) x – 2x – 4y – 4y 18) x2y – x3 – 9y + 9x 19) x2(x – 1) + 16(1– x) 20) 2x2 + 3x – 2xy – 3y 21) x3 – 4x2 + 4x 2 22) 15x y + 20xy − 25xy 23) 4x + 8xy − 3x − 6y 24) x3 + 6x2 + 9x 25) x2 – xy + x – y 26) xy – 2x – y2 + 2y 27) x2 + x – xy – y 2 2 28) x + 4x – y + 29) x – 2xy + y – 30) x2 – 2xy + y2 – x + y 31) xz + yz – 5x – 5y 32) x2 – y2 – 2x – 2y 33) x2 – – 2xy + 2y 2 34) (x + 3) – (2x – 5)(x+ 3) 35) (3x + 2) + (3x – 2)2 – 2(9x2 – 4) Bài Phân tích đa thức thành nhân tử phương pháp dùng đẳng thức: 1) (x + y)2 − 25 2) 100 – (3x – y)2 3) 64x2 – (8a + b)2 4) 4a2b4 – c4d2 5) 7x3 – a3b3 6) 16x3 + 54y3 3 2 7) 8x – y 8) (a + b) – (2a – b) 9) (a + b)3 – (a – b)3 10) (a + b)3 + (a – b)3 11) (6x – 1)2 – (3x + 2) 12) (3x – 1)2 – 16 2 2 13) (5x – 4) – 49x 14) (2x + 5) – (x – 9) 15) (3x + 1)2 – 4(x – 2)2 16) 9(2x + 3)2 – 4(x + 1)2 17) 4b2c2 – (b2 + c2 – a2 )2 18) (ax + by)2 – (ay + bx)2 2 2 2 19) (a + b – 5) – 4(ab + 2) 20) 25 – a + 2ab – b 21) x – y6 2 2 22) x – 4x y + y + 2xy 23) (xy + 1) – (x + y)2 24) x3 – 3x2 + 3x – – y3 25) (x2 – 25)2 – (x – 5)2 26) –4x2 + 12xy – 9y2 + 25 27) x6 – x4 + 2x3 + 2x2 28) (x + y) – – 3xy(x + y – 1) 29) 4(2x – 3)2 – 9(4x2 – 9)2 30) x3 – + 5x – + 3x – 31) (2x + 2)2 + 2(2x+2)(2x – 2) + (2x – 2)2 Bài Phân tích đa thức thành nhân tử phương pháp tách hạng tử thành nhiều hạng tử: 1) x2 + 8x + 15 2) x2 – x – 12 3) x2 – 8x +7 2 4) x – 5x + 5) x – 3x – 6) x2 – 6x + 7) 3x2 + 9x – 30 8) x2 – 9x + 18 9) x2 – 5x – 14 2 10) x – 7x + 12 11) x – 7x + 10 12) x2 + 6x + 13) 3x2 – 5x – 14) 2x2 + x – 15) 7x2 + 50x + 2 16) 12x + 7x – 12 17) 15x + 7x – 18) 2x2 + 5x + 2 19) 4x – 36x – 56 20) 2x + 10x + 21) x2 + 4xy – 21y2 22) 5x2 + 6xy + y2 23) x2 + 2xy – 15y2 24) x2 – 4xy + 10y2 4 25) x + x – 26) x + 4x – 27) x – 19x – 30 28) x3 – 7x – 29) x3 – 5x2 – 14x Chuyên ĐS-8 HKI Quang Duy (093.50.30.798) Bài 10: Phân tích đa thức sau thành nhân tử tổng hợp: 1) x2 – 25 + y2 + 2xy 2) 81x2 – 6yz – 9y2 – z2 3) 3x2 − 6xy + 3y2 4) 2x2 + 2y2 − x2z + z − y2z − 5) x2 − 2xy + y2 − 16 6) x6 − x4 + 2x3 + 2x 2 2 7) x + 2x + – y 8) x + 2xy + y – 9z 9) x3 – 10x2 + 25x – 16xy2 10) 3xy2 – 2xy +12x 11) 5y3 − 10xy2 + 5yx2 − 20y 12) x2 + 2xy + y2 – xz – yz 13) 9x2 + y2 + 6xy 14) – 12x + 6x2 – x3 15).125x3 – 75x2 + 15x – 2 3 16) x – xz – 9y + 3yz 17) x – x – 5x + 12518) x +2x2 – 6x – 27 19) 12x3 + 4x2 – 27x – 20) 4x4 + 4x3 – x2 – x 21) x6 – x4 – 9x3 + 9x2 2 2 22) x – 4x + 8x – 16x + 16 23) 3a – 6ab + 3b – 12c 24) a2 + 2ab + b2 – ac – bc 25) ac – bc – a2 + 2ab – b2 26) x4 + 27) (x – y +5)2 – 2(x – y +5) + 28) x + 64 29) x + x + 30) x8 + x4 + 31) x5 + x + 32) x3 + x2 + 33) x4 + 2x2 – 24 34) x – 2x – 35) x + 4x + 36) 16x – 5x2 – 37) 2x2 + 7x + 38) 2x2 + 3x – 39) x2 – 4x – 2 40) x + x + x + 41) (x + 1) – 4x 42) x3 – 3x2 – 4x + 12 43) x4 – x3 – x2 + 44) (2x + 1)2 – (x – 1)2 45) x4 + 4x2 – 2 46) – x – y + x – y 47) x(x + y) – 5x – 5y 48) x2 – 5x + 5y – y2 2 2 49) x – y – x – y 50) x – y – 2xy + y 51) x2 – y2 + – 4x 52) x2 + xy – 3x – 3y 53) 4x2 + 4x – 9y2 + 54) 5x3 – 5x2y – 10x2 + 10xy 2 2 2 55) 5x – 10xy + 5y – 20z 56) x – z + y – 2xy 57) x – xy – x2z + yz 58) x2 – 2xy – 4z2 + y2 59) 3x2 – 6xy + 3y2 – 12z2 60) x2 – 6xy + 9y2 – 25z2 2 61) (x + x) – 14(x + x)+ 24 62) (x2 + x)2 +4x2 + 4x – 12 63) (x + 1)(x + 2)(x + 3)(x + 4) + 64) (x + 1)(x + 2)(x + 3)(x + 4) – 24 65) (x + 1)(x + 3)(x + 5)(x + 7) + 15 66) (x + 2)(x + 3)(x + 4)(x + 5) – 24 67) x + 2x + 5x + 4x – 12 68) (x2 + x + 1)(x2 + x + 2) – 12 2 69) (x + 8x + 7)(x + 8x + 15) + 15 70) (x2 + 4x + 8)2 + 3x(x2 + 4x + 8) + 2x2 3 3 71) (x+y+x) – x – y – z 72) xy(x + y) + yx(y – z) – zx(z + x) 73) x6 – x4 + 2x3 + 2x2 74) x2(y – z) + y2(z – x) + z2(x – y) 3 75) x + y + z – 3xyz 76) x(x + 4)(x – 4) – (x2 + 1)(x2 – 1) 77) (y – 3)(y + 3)(y2 + 9) – (y2 + 2)(y2 – 2) 78) (a + b – c)2 – (a – c)2 – 2ab + 2bc Bài 11 Tìm x : 1) 5x(x –1) = x – 2) x3–16x =0 3) 3x3 – 27x = 4) 3x – 48x = 5) 36x – 49 = 6) (x – 3)2 – = 7) x2 – 2x = 24 8) 2(x + 5) – x2 – 5x = 9) x3 + x2 – 4x = 10) 5(2x – 1) + 4(8 – 3x) = –5 11).3x(12x – 4) – 9x(4x – 3) = 30 12) x(3x – 2) – 3x(x + 7) = 23 13) 2x(x – 5) – x(3 + 2x) = 26 14) (x – 1)(x+2) –x – = 15) x(2x – 3) – 2(3 – 2x) = 16) 2x(x – 5) – x(3 + 2x) = 26 17) 3x(12x – 4) – 2x(18x +3) = 36 18) 2(x+5) – x2 – 5x = 19) x2(x+1) + 2x(x + 1) = 20) 6x – (2x + 5)(3x – 2) = 21) (2x – 3)2 – (x + 5)2 = 22) (x – 4)2 – (x – 2)(x + 2) = 23) (x + 3)2 – (4 – x)(4 + x) = 10 24) (x – 2) – (x – 3)(x + 3) = 25) (x + 4)2 – (1 – x)(1 + x) = 26) 4(x – 3)2 – (2x – 1)(2x + 1) = 10 27) 9(x + 1)2 – (3x – 2)(3x + 2) = 10 28) 25(x + 3) + (1 – 5x)(1 + 5x) = 29) –4(x – 1)2 + (2x – 1)(2x + 1) = – 30) (x –2)2 – (x + 3)2 – 4(x + 1) = 31) (2x – 3) (2x + 3) – (x – 1)2 – 3x(x – 5) = –44 32) (5x + 1) – (5x + 3) (5x – 3) = 30 33) (x + 3)2 + (x – 2)(x + 2) – 2(x – 1)2 = 34) (12x – 5)(4x – 1) + (3x – 7)(1 – 16x) = 81 Bài 12 Dựa vào đẳng thức để tính nhanh: 1) 252 – 152 2) 2055 – 952 3) 362 – 142 4) 9502 – 8502 5) 972 – 32 6) 412+ 82.59 + 592 7) 892 – 18.89+92 8) 1,242 – 2,48.0,24 + 0,242 Bài 13 Tính giá trị biểu thức sau: a) (x – y)2 b) (x + y)2 c) (x2 – y2)2 3 4 d) x + y e) x + y f) x6 + y6 g) x3 – y3 h) x4 – y4 i) x6 – y6 Chuyên ĐS-8 HKI Quang Duy (093.50.30.798) Cho x + y = 9; xy = 14 Cho x + y = 5; xy = Biết x – y = 6; xy = 16 Bài 14 Chứng minh rằng: a) Nếu: a2 + b2 + c2 = ab + ac + bc b) Nếu: a2 + b2 + c2 + = 2(a + b + c) c) Nếu: 2(a2 + b2) = (a+b)2 d) Nếu a2 + b2 + = ab + a + b a = b = c; a = b = c =1 thì: a = b thì: a=b=1 Bài 15 Chứng minh với số nguyên n biểu thức: a) (2n + 3)2 – chia hết cho với n∈Z b) n (n + 1) + 2n(n + 1) chia hết cho với n∈Z c) n(2n – 3) – 2n(n + 1) chia hết cho với n∈Z d) Biết số tự nhiên n chia cho dư CMR: n2 chia cho dư e) Biết số tự nhiên n chia cho dư CMR: n2 chia cho dư f) Nếu A = 5x + y chia hết cho 19 B = 4x – 3y chia hết cho 19 g) Nếu C = 4x + 3y chia hết cho 13 D = 7x – 2y chia hết cho 13 Bài 16 Chứng minh a) x2 + 2x + > b) x2 + x + > d) – x2 + 4x – < e) 4x – 10 – x2 < 2 g) x + y + 2xy + > f) 4(x–2)(x–1)(x+4)(x+8) + 25x2 ≥ CHIA ĐA THỨC: 1) (–2)5:( –2)3 4) (2x6):(2x)3 7) (x + 2)9 :(x + 2)6 10) 2(x2 + 1)3 : (x2 + 1) 2) (–y)7:( –y)3 5) (–3x)5:(–3x)2 8) (x − y)4 :(x − 2)3 11) 5(x − y)5 : (x − y)2 c) (x – 3)(x – 5) + > f) x2 + 2x + y2 + ≥ 3) (x)12:( –x10) 6) (xy2)4:(xy2)2 9) (x2 + 2x + 4)5 :(x2 + 2x + 4) 12) x3y3 :  − x2y2   ÷   13) 6xy2 :3y 14) 6x2y3 : 2xy2 15) 8x2y : 2xy 16) 5x2y5 : xy3 17) xy3z4 :(−2xz3) 18) (−4x4y3) : 2x2y 19) 9x2y4z :12xy3 20) (2x3y)(3xy2) : 2x3y2 21) (2x3 − x2 + 5x) : x 22) (3x4 − 2x3 + x2):(−2x) 23) (−2x5 + 3x2 – 4x3) : 2x2 24) (x3 – 2x2y + 3xy2): ( −2x) 25) (3a2b)3(ab3)2 26) (2xy2)3(3x2y)2 (a2b2)4 (2x3y2)2 3  3 27)  a6x3 + a3x4 − ax5 ÷: ax3 28) 3(x − y)5 − 2(x − y)4 + 3(x − y)2 :5(x − y)2 10 5  29) (3x5y2 + 4x3y3 − 5x2y4) : 2x2y2 30) (9x2y3 − 15x4y4) :3x2y − (2 − 3x2y)y2 31) (6x2 − xy) : x + (2x3y + 3xy2) : xy − (2x − 1)x 32) (x2 − xy) : x + (6x2y5 − 9x3y4 + 15x4y2): x2y3 CHIA ĐA THỨC CHO ĐA THỨC 1) (x3 – 3x2) :( x – 3) 2) (2x2 + 2x − 4) :(x + 2) 3) (x4 – x – 14) :(x – 2) 4) (x3 − 3x2 + x − 3):(x − 3) Chuyên ĐS-8 HKI Quang Duy (093.50.30.798) 5) (x + x – 12):( x – 2) 6) (2x − 5x + 6x – 15) :(2x – 5) 7) (−3x3 + 5x2 − 9x + 15) :(5− 3x) 8) (− x2 + 6x3 − 26x + 21) :(2x − 3) 9) (2x4 − 5x2 + x3 − − 3x) : (x2 − 3) 10) (x5 + x3 + x2 + 1): (x3 + 1) 11) (2x3 + 5x2 – 2x + 3) :(2x2 – x + 1) 12) (8x − 8x3 − 10x2 + 3x4 − 5) :(3x2 − 2x + 1) 13) (− x3 + 2x4 − − x2 + 7x):(x2 + x − 1) 14) (5x2 + 9xy − 2y2):(x + 2y) 15) (x4 − x3y + x2y2 − xy3) :(x2 + y2) 16) (2a3 + 7ab2 − 7a2b − 2b3) :(2a − b) 17) (4x5 + 3xy4 − y5 + 2x4y − 6x3y2) :(2x3 + y3 − 2xy2) 18) (2x + 4y)2 :(x + 2y) − (9x3 − 12x2 − 3x) :(−3x) − 3(x2 + 3) 19) (13x2y2 − 5x4 + 6y4 − 13x3y − 13xy3):(2y2 − x2 − 3xy) Tìm a, b để đa thức f (x) chia hết cho đa thức g(x) , với: a) f (x) = x4 − 9x3 + 21x2 + ax + b , g(x) = x2 − x − b) f (x) = x4 − x3 + 6x2 − x + a , g(x) = x2 − x + c) f (x) = 3x3 + 10x2 − 5+ a , g(x) = 3x + d) f (x) = x3 – 3x + a , g(x) = (x – 1)2 e) f(x) = x3 + x2 + a – x ; g(x) = (x + 1)2 Thực phép chia f (x) cho g(x) để tìm thương dư: a) f (x) = 4x3 − 3x2 + 1, g(x) = x2 + 2x − b) f (x) = − 4x + 3x4 + 7x2 − 5x3 , g(x) = 1+ x2 − x c) f (x) = 19x2 − 11x3 + − 20x + 2x4 , g(x) = 1+ x2 − 4x d) f (x) = 3x4y − x5 − 3x3y2 + x2y3 − x2y2 + 2xy3 − y4 , g(x) = x3 − x2y + y2 BÀI TẬP ƠN CHƯƠNG Bài 1.Tính: 1) (3x + 4)2 2) (–2x + 1)2 3) (7 – x)2 2 4) (x + 2y) 5) (2x – 1,5) 6) (5 – y)2 7) (x – 5y)(x + 5y) 8) (x – y + 1)(x – y – 1) 9) (x2 – 4)(x2 + 4) 3 10) (x – 3y)(x + 3y) 11) (x – y + z)(x + y + z) 12) (x + – y)(x – – y) 2 13) (a + b + c) 14) (a – b + c) 15) (a – b – c)2 2 16) (x – 2y + 1) 17) (3x + y – 2) 18) (2x – 3y+1)2– (x+3y–1)2 19) (x – y)(x + y)(x2 + y2)(x4 + y4) 21) (a – b + c)2 + 2(a – b + c)(b – c) + (b – c)2 2 22) (x – 3) + 2(x – 3)(x + 3) + (x + 3) 23) (3x – 4y + 7)2 + 8y(3x – 4y + 7) + 16y2 24) (3x3 – 2x2 + x + 2).5x2 25) (a2x3 – 5x + 3a).( – 2a3x) 26) (3x2 + 5x – 2) (2x2 – 4x + 3) 27) (a4 + a3b + a2b2 + ab3 + b4)(a – b) 2 28) (x + x – 1) (x – x + 1) 29) (a + 2)(a – 2)(a2 + 2a + 4)(a2 – 2a + 4) 30) (2 + 3y)2 – (2x – 3y)2 – 12xy31) (x + 1)3 – (x – 1)3 – (x3 – 1) – (x – 1)(x2 + x + 1) Bài Trong biểu thức sau, biểu thức không phụ thuộc vào x: a) (x − 1)3 − (x + 1)3 + 6(x + 1)(x − 1) b) (x + 1)(x2 − x + 1) − (x − 1)(x2 + x + 1) c) (x − 2)2 − ( x − 3)( x − 1) e) (x − 1)3 − (x + 1)3 + 6(x + 1)(x − 1) Bài Tính giá trị biểu thức sau: a) A = a3 − 3a2 + 3a + với a = 11 d) (x + 1)(x2 − x + 1) − (x − 1)(x2 + x + 1) f) (x + 3)2 − (x − 3)2 − 12x b) B = 2(x3 + y3) − 3(x2 + y2) với x + y = Chuyên ĐS-8 HKI Quang Duy (093.50.30.798) Bài Phân tích đa thức sau thành nhân tử: a) 1+ 2xy − x2 − y2 b) a2 + b2 − c2 − d2 − 2ab + 2cd c) a3b3 − d) x2(y − z) + y2(z − x) + z2(x − y) e) x2 − 15x + 36 f) x12 − 3x6y6 + 2y12 g) x8 − 64x2 h) (x2 − 8)2 − 784 Bài Thực phép chia đa thức sau: (đặt phép chia vào bài) a) (35x3 + 41x2 + 13x − 5):(5x − 2) b) (x4 − 6x3 + 16x2 − 22x + 15): (x2 − 2x + 3) c) (x4 − x3y + x2y2 − xy3) :(x2 + y2) d) (4x4 − 14x3y − 24x2y2 − 54y4) :(x2 − 3xy − 9y2) e) (3x4 − 8x3 − 10x2 + 8x − 5):(3x2 − 2x + 1) f) (2x3 − 9x2 + 19x − 15) :( x2 − 3x + 5) g) (15x4 − x3 − x2 + 41x − 70) :(3x2 − 2x + 7) h) (6x5 − 3x4y + 2x3y2 + 4x2y3 − 5xy4 + 2y5) :(3x3 − 2xy2 + y3) Bài Giải phương trình sau: a) x3 − 16x = b) 2x3 − 50x = c) x3 − 4x2 − 9x + 36 = d) 5x2 − 4(x2 − 2x + 1) − = e) (x2 − 9)2 − (x − 3)2 = f) x3 − 3x + = g) (2x − 3)(x + 1) + (4x3 − 6x2 − 6x) :(−2x) = 18 Bài Tìm giá trị lớn giá trị nhỏ biểu thức sau: a) x2 + x + b) 2+ x − x2 c) x2 − 4x + d) 4x2 + 4x + 11 e) 3x2 − 6x + f) x2 − 2x + y2 − 4y + Bài Tính giá trị biểu thức sau cách hợp lý: A = x5 – 20x4 + 20x3 – 20x2 + 20x – x = 99 B = x – 20x – 20x – 20x – 20x – 20x + x = 21 C = x7 – 26x6 + 27x5 – 47x4 – 77x3 + 50x2 + x – 24 x =25 Bài Tính giá trị biểu thức sau cách hợp lý: a) A = (2582 – 2422):(2542 – 2462) b) B = 2632 + 74.263 + 372 c) C = 1362 – 92.136 + 462 d) D = ( 502 + 482 + 462 + …+ 22) – (492 + 472 + 452 + …+ 12) Bài 10 Cho số lẻ liên tiếp CMR hiệu tích hai số cuối với tích hai số đầu chia hết cho 16 Bài 11 Cho b + c = 10 Chứng minh đẳng thức: (10a +b)(10a + c) = 100a(a+1) + bc Áp dụng để tích nhẩm: 62.68; 43.47 Bài 12 Xác định hệ số a, b, c biết rằng: a) (2x – 5)(3x + b) = a2 + x +c b) (ax + b)(x2 – x – 1) = ax3 + cx2 – Bài 13 Cho m số nguyên dương nhỏ 30 Có giá trị m để đa thức: x2 + mx + 72 tích hai đa thức bậc với hệ số nguyên? Bài 14 Phân tích đa thức A thành tích nhị thức bậc với đa thức bậc ba với hệ số nguyên cho hệ số cao cảu đa thức bậc ba 1: 3x4 + 11x3 – 7x2 – 2x + ... 12 xy 31) (x + 1) 3 – (x – 1) 3 – (x3 – 1) – (x – 1) (x2 + x + 1) Bài Trong biểu thức sau, biểu thức không phụ thuộc vào x: a) (x − 1) 3 − (x + 1) 3 + 6(x + 1) (x − 1) b) (x + 1) (x2 − x + 1) − (x − 1) (x2... 2x2 + x – 15 ) 7x2 + 50x + 2 16 ) 12 x + 7x – 12 17 ) 15 x + 7x – 18 ) 2x2 + 5x + 2 19 ) 4x – 36x – 56 20) 2x + 10 x + 21) x2 + 4xy – 21y2 22) 5x2 + 6xy + y2 23) x2 + 2xy – 15 y2 24) x2 – 4xy + 10 y2 4 25)... 12 ) 2(2x – 1) + 3 (1 – 2x) 13 ) 10 x(x – y) – 8y(y – x) 14 ) 3x(y + 2) – 3(y + 2) 15 ) x2 – y2 – 2x + 2y 2 16 ) 2x + 2y – x – xy 17 ) x – 2x – 4y – 4y 18 ) x2y – x3 – 9y + 9x 19 ) x2(x – 1) + 16 (1? ?? x) 20)