Đề bài: Phân tích các đa thức sau thành nhân tử
1) 4x2 – 8xy + 4y2;
2) 5x(x – 1) – 3x2(1 – x);
3) x2 – y2 – 5x + 5y;
4) 3x2 – 6xy + 3y2 – 12z2;
5) 4x2 – y2 + 4x + 1;
6) x5 – 3x4 + 3x3 – x2;
7) –x2 – y2 + 2xy + 36;
8) x3 – x2 – 5x + 125;
9) 6x2 – 5x + 1;
10) x2 – 2x – 9y2 + 6y;
11) (x2 + 1)2 – 4x2;
12) x2 + 2x – 15;
13) x2 – 4xy + 4y2 – z2 + 4zt – 4t2;
14) x3 – x + 3x2y + 3xy2 – y + y3.
Hướng dẫn giải:
1) 4x2 – 8xy + 4y2
= 4(x2 – 2xy + y2)
= 4(x – y)2.
2) 5x(x – 1) – 3x2(1 – x)
= 5x(x – 1) + 3x2(x – 1)
= (5x + 3x2)(x – 1)
= x(5 + 3x)(x – 1).
3) x2 – y2 – 5x + 5y
= (x – y)(x + y) – 5(x – y)
= (x – y)(x + y – 5).
4) 3x2 – 6xy + 3y2 – 12z2
= 3(x2 – 2xy + y2 – 4z2)
= 3[(x – y)2 – (2z)2]
= 3(x – y – 2z)(x – y + 2z).
5) 4x2 – y2 + 4x + 1
= (2x + 1)2 – y2
= (2x + 1 – y)(2x + 1 + y).
6) x5 – 3x4 + 3x3 – x2
= x2(x3 – 3x2 + 3x – 1)
= x2(x – 1)3.
7) –x2 – y2 + 2xy + 36
= 36 – (x2 – 2xy + y2)
= 62 – (x – y)2
= (6 – x + y)(6 + x – y).
8) x3 – x2 – 5x + 125
= (x3 + 125) – (x2 + 5x)
= (x + 5)(x2 – 5x + 25) – x(x + 5)
= (x + 5)(x2 – 5x + 25 – x)
= (x + 5)(x2 – 6x + 25).
9) 6x2 – 5x + 1
= 6x2 – 3x – 2x + 1
= 3x(2x – 1) – (2x – 1)
= (3x – 1)(2x – 1).
10) x2 – 2x – 9y2 + 6y
= (x2 – 9y2) – (2x – 6y)
= (x – 3y)(x + 3y) – 2(x – 3y)
= (x – 3y)(x + 3y – 2).
11) (x2 + 1)2 – 4x2
= (x2 + 1 – 2x)(x2 + 1 + 2x)
= (x – 1)2.(x + 1)2.
12) x2 + 2x – 15
= (x2 – 3x) + (5x – 15)
= x(x – 3) + 5(x – 3)
= (x + 5)(x – 3).
13) x2 – 4xy + 4y2 – z2 + 4zt – 4t2
= (x – 2y)2 – (z – 2t)2
= (x – 2y + z – 2t)(x – 2y – z + 2t).
14) x3 – x + 3x2y + 3xy2 – y + y3
= (x3 + 3x2y + 3xy2 + y3) – (x + y)
= (x + y)3 – (x + y)
= (x + y)[(x + y)2 – 1]
= (x + y)(x + y – 1)(x + y + 1).