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\(=\left(x^3-2x^2+x+2x^2-4x+2-2x+7\right):\left(x^2-2x+1\right)\\ =\left[\left(x^2-2x+1\right)\left(x+2\right)-2x+7\right]:\left(x^2-2x+1\right)\\ =x+2\left(dư:-2x+7\right)\)
1) \(\left(x^2+x+1\right)\left(x^2+x+2\right)-12=x^4+x^3+2x^2+x^3+x^2+2x+x^2+x+2-12\)
\(=x^4+2x^3+4x^2+3x-10=\left(x^4+2x^3\right)+\left(4x^2+8x\right)+\left(-5x-10\right)\)
\(=x^3.\left(x+2\right)+4x.\left(x+2\right)-5.\left(x+2\right)=\left(x+2\right)\left(x^3+4x-5\right)\)
\(=\left(x+2\right)\left(x^3-x^2+x^2-x+5x-5\right)=\left(x+2\right)\left(x-1\right)\left(x^2+x+5\right)\)
2) \(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24=\left[\left(x+2\right)\left(x+5\right)\right].\left[\left(x+3\right)\left(x+4\right)\right]-24\)
\(=\left(x^2+7x+10\right).\left(x^2+7x+12\right)-24\)
Đặt \(a=x^2+7x+10\) thì ta có :\(a.\left(a+2\right)-24=a^2+2a-24=\left(a^2+2a+1\right)-25=\left(a+1\right)^2-5^2\)
\(=\left(a+1+5\right)\left(a+1-5\right)=\left(a+6\right)\left(a-4\right)\)
Thay a , ta có :
\(\left(x^2+7x+10+6\right)\left(x^2+7x+10-4\right)=\left(x^2+7x+16\right).\left(x^2+x+6x+6\right)\)
\(=\left(x^2+7x+16\right)\left(x+1\right)\left(x+6\right)\)
19. 3x2-4x+1
= 3x2-3x-x+1
= (3x2-3x)-(x-1)
= 3x(x-1)-(x-1)
= (3x-1)(x-1)
20.3x2+4x-7
= 3x2+3x-7x-7
= (3x2+3x)-(7x+7)
= 3x(x+1)-7(x-1)
= (3x-7)(x-1)
21.3x2+7x-6
= 3x2+9x-2x-6
= (3x2+9x)-(2x+6)
= 3x(x+3)-2(x+3)
= (3x-2)(x+3)
22.3x2+3x-6
= 3x2+6x-3x-6
=(3x2+6x)-(3x+6)
= 3x(x+2)-3(x+2)
=(3x-3)(x+2)
= 3(x-1)(x+2)
23. 3x2-3x-6
=(3x2-6x)+(3x-6)
=3x(x-2)+3(x-2)
=(3x+3)(x-2)
= 3(x+1)(x-2)
24.6x2-13x+6
= 6x2-9x-4x+6
= (6x2-9x)-(4x-6)
=3x(2x-3)-2(2x-3)
=(3x-2)(2x-3)
25.6x2+13x+6
= 6x2+9x+4x+6
= (6x2+9x)+(4x+6)
=3x(2x+3)+2(2x+3)
=(3x+2)(2x+3)
26. 6x2+15x+6
= (6x2+12x)+(3x+6)
= 6x(x+2)+3(x+2)
=(6x+3)(x+2)
=3(2x+1)(x+2)
27. 6x2-15x+6
= (6x2-12x)-(3x-6)
= 6x(x-2)-3(x-2)
=(6x-3)(x-2)
=3(2x-1)(x-2)
28. 6x2+20x+6
= (6x2+18x)+(2x+6)
= 6x(x+3)+2(x+3)
= (6x+2)(x+3)
= 2(3x+1)(x+3)
29.6x2-20x+6
= (6x2-18x)-(2x-6)
= 6x(x-3)+2(x-3)
= (6x-2)(x-3)
= 2(3x-1)(x-3)
30.6x2+12x+6
= (6x2+6x)+(6x+6)
= 6x(x+1)+6(x+1)
= (6x+6)(x+1)
= 6(x+1)(x+1)
= 6(x+1)2
Ta có : 6x2 - 11x + 3
= 6x2 - 2x - 9x + 3
= (6x2 - 2x) - (9x - 3)
= 2x(3x - 1) - 3(3x - 1)
= (2x - 3)(3x - 1)
\(=x\left(\frac{x^2}{4}+x+1\right)=x\left(\frac{x}{2}+1\right)^2\)
\(a)6x^2y+9xy^2-2-3y=3xy\left(2x+3y\right)-\left(2x+3y\right)=\left(3xy-1\right)\left(2x+3y\right)\)
\(b)x^2-y^2+4-4x=\left(x-2\right)^2-y^2=\left(x+y-2\right)\left(x-y-2\right)\)
\(c)x^6-y^6=\left(x^3-y^3\right)\left(x^3+y^3\right)=\left(x-y\right)\left(x^2+y^2+xy\right)\left(x+y\right)\left(x^2+y^2-xy\right)\)
\(d)4x^2-9y^2+4x+1=\left(2x+1\right)^2-9y^2=\left(2x+3y-1\right)\left(2x-3y+1\right)\)
\(e)x^2-y^2+4x+4=\left(x+2\right)^2-y^2=\left(x+y+2\right)\left(x-y+2\right)\)
b,x2 -y2 +4-4x
=(x2 -4x +4)-y2
=(x-2)2 -y2
=(x-2-y)(x-2+y)
\(x^7+x^5+1\)
\(=x^7+x^6+x^5-x^6-x^5-x^4+x^5+x^4+x^3-x^3-x^2-x+x^2+x+1\)
\(=x^5\left(x^2+x+1\right)-x^4\left(x^2+x+1\right)+x^3\left(x^2+x+1\right)-x\left(x^2+x+1\right)+\left(x^2+x+1\right)\)
\(=\left(x^5-x^4+x^3-x+1\right)\left(x^2+x+1\right)\)
Thank you bạn nhiều nha!