Bài 1. a) E = x² - 2x + y² + 4y + 8

E = ( x² - 2x + 1) + ( y² + 2.2x + 2²) + 3

E = ( x - 1)² + ( y + 2)² + 3

Do : ( x - 1)² lớn hơn hoặc bằng 0 với mọi x

( y + 2)² lớn hơn hoặc bằng 0 với mọi x

Suy ra : ( x - 1)² + 3 lớn hơn hoặc bằng 3 với mọi x

( y + 2)² + 3 lớn hơn hoặc bằng 3 với mọi x

Vậy , E_min = 3 khi và chỉ khi x - 1 =0 -> x = 1

y + 2 =0 -> y = -2

b) F = x² - 4x + y² - 8y + 6

F = x² - 4x + y² - 8y + 4 + 16 - 14

F = ( x² - 2.2x + 2²) + ( y² - 2.4y + 4²) - 14

F = ( x - 2)² + ( y - 4)² - 14

Do : ( x - 2)² lớn hơn hoặc bằng 0 với mọi x

( y - 4)² lớn hơn hoặc bằng 0 với mọi x

Suy ra : ( x - 2)² - 14 lớn hơn hoặc bằng -14 với mọi x

( y - 4)² -14 lớn hơn hoặc bằng -14 với mọi x

Vậy , F_min = -14 khi và chỉ khi x - 2 =0 -> x = 2

y - 4 = 0 -> y = 4

Bài 2 . a) 3x² - 3y² - 2( x - y)²

= 3( x - y)(x + y) - 2( x - y)( x - y)

= (x - y)( 3x + 3y - 2x + 2y)

b) x³ - 4x² - 9x + 36

= x²(x - 4) - 9( x - 4)

= ( x - 4)( x² - 3²)

= ( x - 4)( x - 3)( x + 3)

c) 3x² - 6xy + 3y² - 12z²

= 3( x² - 2xy + y² - 4z²)

= 3[( x - y)² - ( 2z)²]

= 3( x - y - 2z)( x - y + 2z)

d) 5x² - 10xy + 5y² - 20x²

= 5( x² - 2xy + y² - 4x²)

= 5[ ( x - y)² - ( 2x)² ]

= 5( x - y - 2x)( x - y + 2x)

2.

a) 3x² - 3y² - 2(x - y)²

= 3(x² - y²) - 2(x - y)²

= 3(x - y)(x + y) - 2(x - y)²

= [3(x + y) - 2(x - y)](x - y)

= (3x + 3y - 2x + 2y)(x - y)

= (x + 5y)(x - y)

b) x3 - 4x2 - 9x + 36

= x²(x - 4) - 9(x - 4)

= (x² - 9)(x - 4)

c) 3x² - 6xy + 3y² - 12z²

= 3(x² - 2xy + y²) - 3.4z²

= 3(x - y)² - 3(2z)²

= 3[(x - y)² - (2z²)]

= 3(x - y + 2z)(x - y - 2z)

d) 5x2 - 10xy + 5y2 - 20x2

= 5(x² - 2xy + y²) - 5.4x²

= 5(x - y)² - 5(2x)²

= 5(x - y + 2x)(x - y - 2x)

`a,x^3 - 3x^2 + 1 - 3x`

`=x^3 + 1 - 3x^2 - 3x`

`=(x^3 + 1) - 3x(x+1)`

`=(x+1)(x^2 - x + 1) - 3x(x+1)`

`=(x+1)(x^2 - x + 1 - 3x)`

`=(x+1)(x^2 - 4x + 1)`

`b,x^2 + 4x - 2xy - 4y + y^2`

`=(x^2 -2xy + y^2) + (4x-4y)`

`=(x-y)^2 + 4(x-y)`

`=(x-y)(x-y+4)`

`c,3x^2 -6xy + 3y^2 - 12z^2`

`=3(x^2 -2xy +y^2 - 4z^2)`

`=3[(x-y)^2 - (2z)^2]`

`=3(x-y-2z)(x-y+2z)`
 

a: =x^3+1-3x^2-3x

=(x+1)(x^2-x+1)-3x(x+1)

=(x+1)(x^2-x+1-3x)

=(x+1)(x^2-4x+1)

b: =x^2-2xy+y^2+4x-4y

=(x-y)^2+4(x-y)

=(x-y)(x-y+4)

c: =3(x^2-2xy+y^2-4z^2)

=3[(x-y)^2-4z^2]

=3(x-y-2z)(x-y+2z)

a: =x^3+1-3x^2-3x

=(x+1)(x^2-x+1)-3x(x+1)

=(x+1)(x^2-x+1-3x)

=(x+1)(x^2-4x+1)

b: =x^2-2xy+y^2+4x-4y

=(x-y)^2+4(x-y)

=(x-y)(x-y+4)

c: =3(x^2-2xy+y^2-4z^2)

=3[(x-y)^2-4z^2]

=3(x-y-2z)(x-y+2z)

1)

a) => 16x² - 8x + 1 - 8(2x² + 3x - 4x - 6) = 15

=> 16x² - 8x + 1 - 8(2x² - x - 6) = 15

=> 16x² - 8x + 1 - 16x² + 8x + 48 = 15

=> 49 = 15 (?) (vô lí)

=> Không tìm được x thoả mãn

b) (5x - 2)(x - 2) - 4(x - 3) = x² + 3

=> 5x² - 10x - 2x + 4 - 4x + 12 = x² + 3

=> 5x² - 16x + 16 = x² + 3

=> 4x² - 16x + 16 = 3

=> (2x)² - 2.2x.4 + 4² = 3

=> (2x - 4)² = 3

=> \(\left[{}\begin{matrix}2x-4=\sqrt{3}\\2x-4=-\sqrt{3}\end{matrix}\right.\)           \(\Rightarrow\left[{}\begin{matrix}x=\dfrac{4+\sqrt{3}}{2}\\x=\dfrac{4-\sqrt{3}}{2}\end{matrix}\right.\)

Mong bạn xem lại đề bài!

2) 

a) 5x² - 10xy + 5y² - 20z²

= 5(x² - 2xy + y² - 4z²)

= 5[(x - y)² - (2z)²]

= 5(x - y - 2z)(x - y + 2z)

b) a³ - ay - a²x + xy

= a(a² - y) - x(a² - y)

= (a - x)(a² - y)

c) 3x² - 6xy + 3y² - 12z²

= 3(x² - 2xy + y² - 4z²)

= 3[(x - y)² - (2z)²]

= 3(x - y - 2z)(x - y + 2z)

d) x² - 2xy + tx - 2ty

= x(x - 2y) + t(x - 2y)

= (x + t)(x - 2y)

Bài làm

a) 4x - 8y 

<=> 4( x - 2y )

b) 12x( x - 2y ) - 8y( x - 2y )

<=> ( 12x - 8y )( x - 2y )

<=> 4( 3x - 2y )( x - 2y )

c) 2x + 2y - x² - xy

= 2( x + y ) - x( x + y )

= ( x + y )( 2 - x )

d) x² - 4y² 

<=> ( x - 2y )( x + 2y )

e) x³ + x²y - 4x - 4y

<=> x²( x + y ) - 4( x + y )

<=> ( x - 2 )( x + 2 )( x + y )

g) 3x² - 6xy + 3y² - 12x³ 

<=>3( x² - 3xy + y² - 4x³ ) 

# Học tốt #

a)4(x-2y)

b)(x-2y)(12x-8y)

=4(x-2y)(3x-2y)

c)2(x+y)-x(x+y)

=(2-x)(x+y)

d)(x-2y)(x+2y)

e)x²(x+y)-4(x+y)

=(x+y)(x²-4)

=(x+y)(x-2)(x+2)

g)3(x²-2xy+y²-4z³)

=3[(x-y)²-4z³]

????????????phải là 4z²chứ nhỉ.....

a)  4x-8y =4(x-2y)

b)  =(x-2y)(12x-8y)=(x-2y)4(3x-2y)

c)  =2(x+y) -x(x+y) =(x+y) (2-x) 

d)  =x²-(2y)²=(x-2y)(x+2y)

e)  =x²(x+y) -4(x+y) =(x+y) (x²-4)

=(x+y) (x-2)(x+2)

\(a,=x\left(x-2\right)^2\\ b,=\left(x-y\right)^2-9=\left(x-y-3\right)\left(x-y+3\right)\\ c,=x^2\left(2x-1\right)-4\left(2x-1\right)=\left(x-2\right)\left(x+2\right)\left(2x-1\right)\\ d,=\left(x-y\right)\left(x+y\right)-5\left(x-y\right)=\left(x-y\right)\left(x+y-5\right)\\ e,=3\left[\left(x-y\right)^2-4z^2\right]=3\left(x-y-2z\right)\left(x-y+2z\right)\\ f,=x\left[\left(x-2\right)^2-y^2\right]=x\left(x-y-2\right)\left(x+y-2\right)\\ g,=x\left[\left(x-y\right)^2-25\right]=x\left(x-y-5\right)\left(x-y+5\right)\\ h,=x^3-x-2x+2=x\left(x-1\right)\left(x+1\right)-2\left(x-1\right)\\ =\left(x-1\right)\left(x^2+x-2\right)=\left(x-1\right)^2\left(x+2\right)\\ i,=3x^2+3x-10x-10=\left(x+1\right)\left(3x-10\right)\)

=\(\left(\sqrt{3}x-\sqrt{3}y\right)^2-12z=\left(\sqrt{3}x-\sqrt{3}y-\sqrt{12z}\right)\left(\sqrt{3}x-\sqrt{3}y+\sqrt{12z}\right)\)

x² - x - y² - y

= (x - y)(x + y) - (x + y)

= (x + y)(x - y - 1)

***

9x² + y² - 16z² + 6xy

= (3x + y)² - (4z)²

= (3x + y - 4z)(3x + y + 4z)

***

a³ - a²x - ay + xy

= a²(a - x) - y(a - x)

= (a - x)(a² - y)

***

2x² - 8y² + 3x + 6y

= 2(x² - 4y²) + 3(x + 2y)

= 2(x - 2y)(x + 2y) + 3(x + 2y)

= (x + 2y)(2x - 4y + 3)

***

xy(x + y) + yz(y + z) + xz(x + z) + 2xyz

= xy(x + y + z) + yz(x + y + z) + xz(x + z)

= y(x + y + z)(x + z) + xz(x + z)

= (x + z)(xy + y² + yz + xz)

= (x + z)[y(x + y) + z(x + y)]

= (x + z)(x + y)(y + z)