De cuong on thi

PHÂN TÍCH ĐA THỨC THÀNH NHÂN TƯ
A. PHẦN CƠ BẢN

Bài 1: phân tích các đa thức sau thành nhân tử bằng phương pháp đặt nhân tử chung:
1) xy – 12x – 18y
11) 2mx – 4m2xy + 6mx
21) ab(x–5) –a2(5–x)
2) 8xy – 24xy + 16x
12) 7x2y5 – 14x3y4 – 21y3
22) 2a2(x –y) –4a(y–x)
3) xy – x
13) 2(x–y) – a(x–y)
23) a(x–3) – a2(3–x)
2
4) – ax – ax – a
14) 2a(x+y) – 4(x+y)
24) 5x2y(x – 7) – 5xy(7 – x)
5) – 2ax + 4ay
15) 3a(x –y) –6ab(x–y)
25) 2xy(a–1) – 4x2y(1–a)
6) 2a2b – 4ab2 – 6ab
16) m(a–b) – m2(b–a)
26) 4a(x–3) – 2(3–x)
7) 5ax – 15ay + 20
17) mx(a+b) –m(a+b)
27) xm+1 – xm
2
2
8) 3a x – 6a y + 12a
18) x(a–b) –y(b–a)
28) xm+1 + xm

9) 2axy – 4a2xy2 + 6a3x2
19) a(x–1) + b(1–x)
29) xm+2 – xm
10) 5a2xy – 10a3x – 15ay
20) 2a(x+2) + a2 (–x–2)
30) xm+2 – xm+1
Bài 2: Phân tích các đa thức sau thành nhân tử bằng phương pháp dùng hằng đẳng thức:
1) a2 – 4b2
14) (a+3b)2 – 9b2
27) 4x4 – 16x2y3 + 16y6
2) 4x2 – 9y2
15) (a – 5b)2 – 16b2
28) 9x4 – 12x5 + 4x6
3) 25a4 – 1
16) 36a2 – (3a – 2b)2
29) x10 – 4x8 + 4x6
4
2
2
4) a – 81
17) 4a – (a+b)
30) 8x6 – 27y3
5) 121 – 36x2
18) 49a2 – (2a – b)2
31) (a+b)3 – c3
6) 144a2 – 81
19) 81a2 – (5a – 3b)2
32) x3 – (y – 1)3
7) 36a2 – 49b2
20) (a – 2b)2 – (3a + b)2

33) x6 – 1 ; x6 + 1
4
2
2
2
8) a – 4b
21) (5a – b) – (2a + 3b)
34) x6 – y6 ; x12 – y 4
1
1 2
1 3
22) (2a – b)2 – 4(a – b)2
 a3
x 8
a  b2
35) 27
;8
9) 4
4 4 25
23) 9(a+b)2 – 4(a – 2b)2
36) 27 – 27m + 9m2 – m3
a 
4
10) 9
4
11) a – 16b4
24) 25x2 – 20xy + 4y2
37) 125 – (x+2)3 ; (x+3)3 – 8
12) (a – b)2 – c2
25) 4x2 – 12xy + 9y2

38) (x+4)3 – 64 ; (x+1)3 – 125
2
2
4
2
2
13) (a – 2b) – 4b
26) 9x – 12x y + 4y
39) x9 – 1 ; x9 + 1
Bài 3: Phân tích các đa thức sau thành nhân tử bằng phương pháp nhóm các hạng tử:
1) x(a – b) + a – b
11) x2 – xy + 2x – 2y
21) 3ax2 +2bx2+ax+bx+5a+5b
2) m(x+y) + x + y
12) 10ax – 5ay – 2x + y
22) ax2 – bx2–2ax+2bx–3a+3b
3) x – y – a(x – y)
13) 5x2y + 5xy2 – a2x + a2y
23) 2ax2– bx2 –2ax+bx + 4a –2b
4) a – b – x(b – a)
14) 6a2y – 3aby + 4a2x – 2abx
24) ax2 –5x2 – ax + 5x + a – 5
5) x(a+b) – a – b
15) 2x2 – 6xy + 5x – 15y
25) ax – bx + cx –3a + 3b –3c
2
6) a(x – y) – x + y
16) ax – 3axy + bx – 3by
26) 2ax – bx + 3cx – 2a + b –3c
7) ax + ay – 2x – 2y

17) 2a2x – 5by – 5a2y + 2bx
27) ax –bx –2cx –2a + 2b + 4c
2
3
2
8) x + xy – 2x – 2y
18) 2ax + 6ax + 6ax + 18a
28) 12x2 –3xy + 8xz – 2yz
9) x2 – xy – 4x + 4y
19) ax2y – bx2y – ax+bx+2x – 2b 29) x3 + x2y – x2z – xyz
10) ax – 2by – 2bx + by
20) 10ax – 5ay + 2x – y
30) 2a2c2 – 2abc + bd – acd
Bài 4: Phân tích các đa thức sau thành nhân tử bằng cách phối hợp nhiều phương pháp:
1) 5x2 + 10xy + 5y2
13) 4a2 – x2 – 2x – 1
25) 3x5y3 + 3x2y6
2
2
2
2
2) 6x – 12xy + 6y
14) 36x – a + 10a – 25
26) 2(a+b)3 + 16
3) 2x3 + 4x2y + 2xy2
15) 25a2b2 – 4x2 + 4x – 1
27) ab(x – y)3 + 8ab
4) 4x5y2 + 8x4y3 + 4x3y4
16) x2 – 2x + 1 – a2 – 2ab – b2
28) 8xy3+ x(x – y)3

5) 2x3 + 8x2 + 8x
17) 1 – 2m + m2 – x2 – 4x – 4
29) (a+b)3 + c3
4 2
3 2
2 2
2
2
2
6) 5x y + 20x y + 20x y
18) m – 6m + 9 – x + 4xy–4y 30) x2 + (a+b)xy + aby2
7) (a2 + 4)2 – 16a2
19) x2 +4xy+4y2 – a2 + 2ab– b2 31) x2 – (2a+b)xy + 2aby2
8) (a2 + 9)2 – 36a2
20) 9xy – 4a2xy
32) y2 – (3b+2a)xy + 6abx2
2
2
2 2
3
9) (a + 4b) – 16a b
21) 2xm – 2x
33) 3xy(a2 + b2) – ab(x2 + 9y2)
10) 36a2 – (a2 + 25)2
22) 8a3x – 27b3x
34) (xy + ab)2 + (ay – bx)2


11) x2 + 2xy + y2 – 25
12) x2 – 4xy + 4y2 – 36z2

23) 16a3xy – 54b3xy4
24) 2x3 + 16

35) ab(a2 + b2) + xy(a2 + b2)
36) (xy – ab)2 + (bx – ay)2

B. PHẦN NÂNG CAO
Bài 1: Phân tích các đa thức sau thành nhân tử (bằng phương pháp tách hạng tử, bổ sung hằng

thức, nhẩm nghiệm …)
1) x2 – 5x + 6
15) 4x2 + 15x + 9
29) 4x2 + 8x – 5
2
2
2) x – 7x + 12
16) 3x + 10x + 3
30) x2 – 10x + 24
3) x2 + 5x + 6
17) 5x2 + 14x – 3
31) x2 – 13x + 36
4) x2 +7x + 12
18) 5x2 – 18x – 8
32) x2 – 2x – 8
2
2
5) x – x – 12
19) 6x + 7x – 3
33) x2 – 2x – 3

6) x2 + x – 12
20) 3x2 – 3x – 6
34) x2 + 3x – 18
7) x2 + 9x + 20
21) 3x2 + 3x – 6
35) x2 + 5x – 36
8) x2 + x – 20
22) 6x2 – 13x + 6
36) x2 – 5x – 24
2
2
9) x – 9x + 20
23) 6x + 15x + 6
37) x4 + 3x2 + 4 ; x4 + 5x2 + 9
10) x2 – x – 20
24) 6x2 – 20x + 6
38) x4 – 3x2 + 9
11) 2x2 – 3x – 2
25) – 8x2 + 5x + 3
39) x4 – 7x2 + 1
12) 3x2 + x – 2
26) 8x2 – 10x – 3
40) 2x4 – 21x2 + 1
2
2
13) 4x – 7x – 2
27) 3x + 7x – 6
41) 2x4 – x2 – 1
14) 4x2 + 5x – 6
28) 8x2 + 10x – 3

42) 4x4 – 12x2 + 1
Bài 2: Phân tích các đa thức sau thành nhân tử:
1) x2 – xy – 2y2
9) 3x2 + 8xy – 3y2
17) 6a2 – ab – 2b2 + a + 4b – 2
2) x2 + xy – 2y2
10) x2 – x – xy – 2y2 + 2y
18) 2x2+5x –12y2+12y–3– 10xy
3) x2 – 3xy + 2y2
10) x2 +2y2– 3xy + x – 2y
19) 2a2 + 5ab – 3b2 – 7b – 2
4) x2 – xy – 6y2
11) x2 +x – xy – 2y2 + y
20) 2x2 – 7xy + x + 3y2 – 3y
2
2
2
2
5) 2x – 3xy – 6y
12) x – 4xy – x + 3y + 3y
21) 6x2 – xy – 2y2+3x – 2y
6) 3x2 – 5xy – 2y2
13) x2 + 4xy + 2x + 3y2 + 6y
22) 4x2 – 4xy – 3y2 – 2x + 3y
7) 6x2 + 2xy – 4y2
14) 6x2 +xy – 7x – 2y2 + 7y – 5 23) 2x2 – 3xy– 4x – 9y2 – 6y
8) 2x2 + 2xy – 4y2
15) 3x2 – 22xy –4x + 8y+7y2+1 24) 3x2 – 5xy + 2y2 – 4x – 4y
Bài 3: Phân tích các đa thức sau thành nhân tử bằng cách thêm bớt cùng một hạng tử để tạo
HĐT

1) 4x4 + 1
5) x8 + 4
9) 4x4y4 + 1
2) 4x8 + 1
6) x4 + 182
10) 64x4 + y4
3) x4 + 64
7) 4x4 + 81
11) x4 + 324
4
8
4) x + 4
8) x + 64
12) x12 + 4
Bài 4: Phân tích các đa thức sau thành nhân tử (dạng thêm bớt khi số mũ chia 3 dư 1, chia cho 3 dư 2)

1) x4 + x2 + 1
2) x5 + x + 1
3) x7 + x5 + 1
4) x5+ x – 1
5) x5 – x4 – 1
6) x7 + x2 + 1
7) x10 + x5 + 1
8) x7 + x5 – 1

9) x8 + x + 1
10) x5 + x4 + 1
11) x 11 + x + 1
12) x8 + x7 + 1
13) x10 + x8 + 1

14) x11 + x4 + 1
15) x11 + x10 + 1
16) x11 + x7 + 1

17) x4 + 2014x2 + 2013x + 2014
18) x4 + 2014x2 – 2015x + 2014
19) x4 + 2016x2 + 2015x + 2016
20) x4 + 2012x2 – 2011x + 2012
21*) x12 + x6 + 1
22) x16 + x8 + 1
23) x8 + 7x4 + 16
24) x40 + 2x20 + 9

Bài 5: Phân tích các hằng đẳng thức sau thành nhân tử (Sử dụng HĐT tổng, hiệu hai lập phương)

1) a3 + b3 + c3 – 3abc
6) 8a3 + 27b3 + 64c3 – 72abc
11) (x+y – z)3 – x3 – y3 + z3
2) x3 – y3 + z3 + 3xyz
7) 27a3 + 64b3 + 125c3 – 180abc
12) (x – y)3+(y+z)3 – (x – y)3
3) a3 + b3 – c3 + 3abc
8) 125a3 +8b3+27c3 – 90abc
13) (x – y – z)3 – x3 + y3 + z3
4) x3 – y3 – z3 – 3xyz
9) 64x3 +125y3 +216z3 – 360xyz
14) (x+2y+3z)3 –x3 – 8y3 – 27z3
3
3
3

3
3
3
3
5) x + 8y + 27z – 18xyz
10) (x+y+z) – x – y – z
15) (a – b)3 + (b – c)3 + (c – a)3
Bài 6: Phân tích các đa thức sau thành nhân tử (bằng phương pháp đặt ẩn phụ)


1) 6x4 – 11x2 + 3
9) (x2 – x)2 + 3(x2 – x) + 2
17) x2 +6xy + 9y2 – 3x – 9y +2
2) x4 + 3x2 – 4
10) (x2+3x)2 + 7x2+21x+10
18) x2–4xy +4y2 –7x + 14y +6
3) 3x4 + 4x2 + 1
11) (x2+5x)2 +10x2+50x+24
19) 4x2 +4xy +y2 + 10x + 5y– 6
4) x4 + x2 – 20
12) (5x2 – 2x)2 + 2x – 5x2 – 6
20) (x2+x+1)(x2+x+2) – 12
2
2
2
2
2
5) (x +x) + 4(x +x) – 12
13) x +2xy+y +2x+2y – 15
21) (x2–x+3)(x2–x–2) + 4

6) (x2+x)2 + 9x2 + 9x + 14
14) x2 +8xy + 16y2+2x+8y – 3
22) (2x2+x–2)(2x2+x–3) – 12
7) (x – y)2 + 4x – 4y – 12
15) x2 + 2xy + y2 – x – y – 12
23) (2x2–x–1)(2x2–x–4) –10
8) (x2+x) – 2(x2+x) – 15
16) x2 –4xy+4y2 –2(x – 2y)– 35 24) (x2+3x+3)(x2+3x+5) – 80
Bài 7: Phân tích đa thức thành nhân tử bằng phương pháp Đặt ẩn phụ
dạng (x+a)(x+b)(x+c)(x+d) + e với (a+b = c+d)
1) (x+2)(x+4)(x+6)(x+8) + 16
10) (x – 7)(x – 5)(x – 4)(x – 2) – 72
2) (x+2)(x+3)(x+4)(x+5) – 24
11) (x2 + 8x + 12)(x2 + 12x + 32) + 16
3) x(x+4)(x+6)(x+10) + 128
12) (x2 + 6x + 8)(x2 + 8x + 15) – 24
4) (x+1)(x+2)(x+3)(x+4) – 24
13) (x2 + 4x + 3)(x2 + 6x + 8) – 24
5) x(x+1)(x+2)(x+3) +1
14) (x2 – 6x + 5)(x2 – 10x + 21) – 20
6) (x – 1)(x – 3)(x – 5)(x – 7) – 20
15) (x2 + x – 2)(x2 + 9x + 18) – 28
7) (x – 1)(x + 2)(x + 3)(x + 6) – 28
16) (x2 + 5x + 6)(x2 – 15x + 56) – 144
8) x(x – 1)(x + 1)(x + 2) – 3
17) (x2 – 11x + 28)(x2 – 7x + 10) – 72
9) (x + 2)(x + 3)(x – 7)(x – 8) – 144
18) (x2 – 2x)(x2 – 4x + 3) + 1
Bài 8: Phân tích đa thức thành nhân tử bằng phương pháp đặt ẩn phụ dạng đẳng cấp.
1) (x2 + 1)2 + 3x(x2 + 1) + 2x2

6) (x2 – x + 2)4 – 3x2(x2 – x + 2)2 + 2x4
2) (x2 – 1)2 – x(x2 – 1) – 2x2
7) 3(– x2+2x+3)4 – 26x2(–x2+2x+3)2 – 9x4
3) (x2 + 4x + 8)2 + 3x(x2 + 4x+8) + 2x2
8) –6(–x2–x+1)4 +x2(–x2–x+1)2 + 5x4
4) 4(x2 +x+1)2 + 5x(x2 + x+1) + x2
9) (x2 –x–1)4 + 7x2(x2–x–1)2 + 12x4
2
2
2
2
5) (x – x +1) – 5x(x – x + 1) + 4x
10) 10(x2 – 2x +3)4 –9x2(x2 –2x +3)2 – x4

Bài 9: Phân tích đa thức thành nhân tử bằng phương pháp đặt ẩn phụ dạng hồi quy.
1) x4 + 6x3 + 11x2 + 6x + 1
6) x4 – 10x3 + 26x2 – 10x + 1
2) x4 + x3 – 4x2 + x + 1
7) x4 + 7x3 + 14x2 + 14x + 4
4
3
2
3) x + 6x + 7x + 6x + 1
8) x4 – 10x3 – 15x2 + 20x + 4
4) x4 + 5x3 – 12x2 + 5x + 1
9) 2x4 – 5x3 – 27x2 + 25x + 50
5) 6x4 + 5x3 –38x2 + 5x + 6
10) 3x4 + 6x3 – 33x2 – 24x + 48
Bài 10: Phân tích đa thức thành nhân tử bằng phương pháp đặt ẩn phụ
Dạng (x+a)(x+b)(x+c)(x+d) + ex2 (với ad = bc)

1) (x –3)(x – 5)(x – 6)(x – 10) – 24x2
6) (x – 2)(x – 4)(x – 5)(x – 10) – 54x2
2) (x – 1)(x + 2)(x + 3)(x – 6) + 32x2
7) (x+2)(x – 4)(x + 6)(x – 12) + 36x2
2
3) (x + 1)(x – 4)(x + 2)(x – 8) + 4x
8) 4(x+5)(x+6)(x+10)(x+12) – 3x2
4) (x – 2)(x – 3)(x – 6)(x – 4) – 72x2
9) (x+2)(x+3)(x+8)(x+12) – 4x2
5) (x + 3)(x – 1)(x – 5)(x + 15) + 64x2
10) (x – 18)(x – 7)(x +35)(x + 90) – 67x2

Bài 11: Phân tích đa thức thành nhân tử bằng phương pháp đoán nghiệm (dùng MTBT hỗ trợ).
1) x3 – 6x2 + 11x – 6
2) x3 – x2 – 4x + 4
3) 2x3 + 3x2 – 8x + 3
4) – 6x3 + x2 + 5x – 2
5) x3 – 5x2 + 2x + 8
6) 3x3 + 19x2 + 4x – 12
7) x3 + 3x2 – 10x – 24
8) 2x3 – 11x2 + 10x + 8
9) 2x3 + 11x2 + 3x – 36
10) 6x3 – 17x2 – 4x + 3
11) x4 – 8x3 + 11x2 + 8x – 12
12) – 3x4 + 20x3 – 35x2 – 10x + 48

14) x5 – 5x4 – 2x3 + 17x2 – 13x + 2
15) x5 – 5x4 + 6x3 – x2 + 5x – 6
16) 3x3 – 5x2 + 5x – 2
17) 2x3 + 5x2 + 5x + 3

18) 3x3 + 5x2 – x – 2
19) 4x3 + x2 + x – 3
20) 4x3 – x2 + x + 3
21) 4x3 – 7x2 – x + 3
22) 6x3 – 7x2 + 5x – 2
23) 4x3 – 5x2 + 6x + 9
24) 4x3 + 5x2 + 10x – 12
25) 5x3 – 12x2 + 14x – 4


13) – 2x4 – 7x3 – x2 + 7x + 3
Bài 12: Phân tích đa thức thành nhân tử bằng phương pháp đặt ẩn phụ
1) (2x+1)(x+1)2(2x+3) – 18
9) (2x – 1)(x – 1)(x – 3)(2x + 3) + 9
2
2) (6x+5) (3x+2)(x+1) – 35
10) (4x +1)(12x – 1)(3x + 2)(x + 1) – 4
3) (12x+7)2(3x+2)(2x+1) – 3
11) (x + 1)(2x – 1)(3x + 2)(6x – 5) – 4
2
4) (6x+7) (3x+4)(x+1) – 6
12) (2x – 1)(x + 1)(4x + 3)(8x – 6) – 2
2
5) (2x+1) (4x+1)(4x+3) – 18
13) (2x+1)(4x – 1)(12x – 5) – 4
2
6) (x – 2) (2x – 5)(2x – 3) – 5
14) (4x +1)(2x – 3)(4x – 3)(8x +8) – 130
7) (3x – 2)2(6x – 5)(6x – 3) – 5
15) (4x – 2)(10x + 4)(5x + 7)(2x + 1) + 17

2
8) (x+3) (3x+8)(3x+10) – 8