d.

$(\frac{x}{2}-y)^3=(\frac{x}{2})^3-3(\frac{x}{2})^2y+3.\frac{x}{2}y^2-y^3$

$=\frac{x^3}{8}-\frac{3x^2y}{4}+\frac{3xy^2}{2}-y^3$

e.

$(\frac{x}{2}+\frac{y}{3})^3=(\frac{x}{2})^3+3(\frac{x}{2})^2\frac{y}{3}+3.\frac{x}{2}(\frac{y}{3})^2+(\frac{y}{3})^3$

$=\frac{x^3}{8}+\frac{x^2y}{4}+\frac{xy^2}{6}+\frac{y^3}{27}$

f.

$(\frac{2x}{3}-2y)^3=(\frac{2x}{3})^3-3(\frac{2x}{3})^2.2y+3.\frac{2x}{3}(2y)^2-(2y)^3$

$=\frac{8x^3}{27}-\frac{8x^2y}{3}+8xy^2-8y^3$

g.

$(x+y)^3+(x-y)^3=(x^3+3x^2y+3xy^2+y^3)+(x^3-3x^2y+3xy^2-y^3)$

$=2x^3+6xy^2$

Lời giải:

a.
$(3-y)^3=3^3-3.3^2y+3.3y^2-Y63=27-27y+9y^2-y^3$
b.

$(3x+2y^2)^3=(3x)^3+3.(3x)^2(2y^2)+3.3x(2y^2)^2+(2y^2)^3$
$=8y^6+24xy^4+24x^2y^2+8x^3$

c.

$(x-3y^2)^3=x^3-3x^2(3y^2)+3x(3y^2)^2-(3y^2)^3$
$=x^3-9x^2y^2+27xy^4-27y^6$

\(a,-x^3+3x^2-3x+1=-\left(x^3-3x^2+3x-1\right)=-\left(x-1\right)^3\)

\(b,8-12x+6x^2-x^3=2^3-3.2^2x+3.2x^2-x^3=\left(2-x\right)^3\)

\(c,x^3-6x^2y+12xy^2-8y^3=x^3-3.2y.x^2+2.\left(2y\right)^2x-\left(2y\right)^3=\left(x-2y\right)^3\)

\(d,8x^3+12x^2+6x+1\\ =\left(2x\right)^3+3.1\left(2x\right)^2+3.2x.1^2+1^3=\left(2x+1\right)^3\)

\(c,x^3+x^2+\dfrac{x}{3}+\dfrac{1}{27}=x^3+3.x^2.\dfrac{1}{3}+3.x.\left(\dfrac{1}{3}\right)^2+\left(\dfrac{1}{3}\right)^3=\left(x+\dfrac{1}{3}\right)^3\\ f,x^3+\dfrac{3}{2}x^2+\dfrac{3}{4}x+\dfrac{1}{8}=x^3+3.x^2.\dfrac{1}{2}=3.x.\left(\dfrac{1}{2}\right)^2+\left(\dfrac{1}{2}\right)^3=\left(x+\dfrac{1}{2}\right)^3\)

a) (x - 2)³ - x(x + 1)(x - 1) + 6x(x + 3)

= x³ - 6x² + 12x - 8 - x(x² - 1) + 6x² + 18x

= x³ + 30x - 8 - x³ + 8

= 30x

b) (x - 2)(x² - 2x + 4)(x + 2)(x² + 2x + 4)

= [(x - 2)(x² + 2x + 4)][(x + 2)(x² - 2x + 4)]

= (x³ - 2³)(x³ + 2³)

= x⁶ - 2⁶

= x⁶ - 64

c) (2x + y)(4x² - 2xy + y²) - (2x - y)(4x² + 2xy + y²)

= [(2x)³ + y³] - [(2x)³ - y³]

= 8x³ + y³ - 8x³ + y³

= 2y³

d) (x + y)³ - (x - y)³ - 2y³

= x³ + 3x²y + 3xy² + y³ - x³ + 3x²y - 3xy² + y³ - 2y³

= 6x²y

e) (x + y + z)² - 2(x + y + z)(x + y) + (x + y)

= x² + y² + z² + 2xy + 2xz + 2yz - 2x² - 2xy - 2xy - 2y² - 2xz - 2yz + x + y

= -x² - y² + z² + x + y

a: =x^3-6x^2+12x-8+6x^2-18x-x(x^2-1)

=x^3-6x-8-x^3+x

=-5x-8

b: =(x-2)(x^2+2x+4)(x+2)(x^2-2x+4)

=(x^3-8)(x^3+8)=x^6-64

c: =8x^3+y^3-8x^3+y^3=2y^3

d: =x^3+3x^2y+3xy^2+y^3-x^3+3x^2y-3xy^2+y^3-2y^3

=6x^2y

e: =(x+y+z)(x+y+z-2x-2y)+(x+y)

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

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

=z^2-x^2-2xy-y^2+x+y

a: =6y^3-3y^2-y^2+2y-y+y^2-y^3

=5y^3-3y^2+y

b: =2x^2a-a-2x^2a-a-x^2-ax

=-x^2-ax-2a

c: =2p^3-p^3+1+2p^3+6p^2-3p^5

=3p^3+6p^2-3p^5+1

d: =-3a^3+5a^2+4a^3-4a^2=a^3+a^2

a) \(\left(x^2-25\right)^2-\left(x-5\right)^2\)

\(=\left(x-5\right)^2\left(x+5\right)^2-\left(x-5\right)^2\)

\(=\left(x-5\right)^2\left[\left(x+5\right)^2-1\right]\)

\(=\left(x-5\right)^2\left(x^2+10x+25-1\right)\)

\(=\left(x-5\right)^2\left(x^2+10x+24\right)\)

b) \(\left(4x^2-25\right)^2-9\left(2x-5\right)^2\)

\(=\left(2x-5\right)^2\left(2x+5\right)^2-9\left(2x-5\right)^2\)

\(=\left(2x-5\right)^2\left[\left(2x+5\right)^2-9\right]\)

\(=\left(2x-5\right)^2\left(4x^2+20x+25-9\right)\)

\(=\left(2x-5\right)^2\left(4x^2+20x+16\right)\)

c) \(4\left(2x-3\right)^2-9\left(4x^2-9\right)^2\)

\(=4\left(2x-3\right)^2-9\left(2x-3\right)^2\left(2x+3\right)^2\)

\(=\left(2x-3\right)^2\left[4-9\left(2x+3\right)^2\right]\)

\(=\left(2x-3\right)^2\left[4-9\left(4x^2+12x+9\right)\right]\)

\(=\left(2x-3\right)^2\left(4-36x^2-108x-81\right)\)

\(=(2x-3)^2(-36x^2-108x-77)\)

d) \(a^6-a^4+2a^3+2a^2\)

\(=\left(a^6-a^4\right)+\left(2a^3+2a^2\right)\)

\(=a^4\left(a^2-1\right)+2a^2\left(a-1\right)\)

\(=a^4\left(a+1\right)\left(a-1\right)+2a^2\left(a-1\right)\)

\(=\left(a-1\right)\left[a^4\left(a+1\right)+2a^2\right]\)

\(=a^2\left(a-1\right)\left[a^2\left(a+1\right)+2\right]\)

\(=a^2\left(a-1\right)\left(a^3+a^2+2\right)\)

e) \(\left(3x^2+3x+2\right)^2-\left(3x^2+2x-2\right)^2\)

\(=\left[\left(3x^2+3x+2\right)-\left(3x^2+3x-2\right)\right]\left[\left(3x^2+3x+2\right)+\left(3x^2+3x-2\right)\right]\)

\(=\left(3x^2+3x+2-3x^2-3x+2\right)\left(3x^2+3x+2+3x^2+3x-2\right)\)

\(=4\left(6x^2+6x\right)\)

\(=4\cdot6x\left(x+1\right)\)

\(=24x\left(x+1\right)\)

d) \(\left(3x+1\right)^2-4\left(x-2\right)^2\)

\(=\left[\left(3x+1\right)-2\left(x-2\right)\right]\left[\left(3x+1\right)+2\left(x-2\right)\right]\)

\(=\left(3x+1-2x+4\right)\left(3x+1+2x-4\right)\)

\(=\left(x+5\right)\left(5x-3\right)\)

c) \(9\left(2x+3\right)^2-4\left(x+1\right)^2\)

\(=\left[3\left(2x+3\right)-2\left(x+1\right)\right]\left[3\left(2x+3\right)+2\left(x+1\right)\right]\)

\(=\left(6x+9-2x-2\right)\left(6x+9+2x+2\right)\)

\(=\left(4x-7\right)\left(8x+11\right)\)

f) \(4b^2c^2-\left(b^2+c^2-a^2\right)^2\)

\(=\left[2bc-\left(b^2+c^2-a^2\right)\right]\left[2bc+\left(b^2+c^2-a^2\right)\right]\)

\(=\left(2bc-b^2-c^2+a^2\right)\left(2bc+b^2+c^2-a^2\right)\)

g: =(ax+by-ay-bx)(ax+by+ay+bx)

=[a(x-y)-b(x-y)]*[a(x+y)+b(x+y)]

=(x-y)(x+y)(a-b)(a+b)

h: =(a^2+b^2-5-2ab-4)(a^2+b^2-5+2ab+4)

=[(a-b)^2-9][(a+b)^2-1]

=(a-b-3)(a-b+3)(a+b-1)(a+b+1)

i: =(4x^2-3x-18-4x^2-3x)(4x^2-3x-18+4x^2+3x)

=(-6x-18)(8x^2-18)

=-12(x+3)(4x^2-9)

=-12(x+3)(2x-3)(2x+3)

k: =(3x+3y-3)^2-(4x+6y+2)^2

=(3x+3y-3-4x-6y-2)(3x+3y-3+4x+6y+2)

=(-x-3y-5)(7x+9y-1)

i: =25-(2x-3y)^2

=(5-2x+3y)(5+2x-3y)

m: =(x-y)^2-(2m-n)^2

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

a) \(x^3+6x^2+12x+8\)

\(=x^3+3\cdot x^2\cdot2+3\cdot2^2\cdot x+2^3\)

\(=\left(x+2\right)^3\)

b) \(x^3-3x^2+3x-1\)

\(=x^3-3\cdot x^2\cdot1+3\cdot x\cdot1^2-1^3\)

\(=\left(x-1\right)^3\)

c) \(1-9x+27x^2-27x^3\)

\(=1^3-3\cdot3x\cdot1^2+3\cdot\left(3x\right)^2\cdot1-\left(3x\right)^3\)

\(=\left(1-3x\right)^3\)

d) \(x^3+\dfrac{3}{2}x^2+\dfrac{3}{4}x+\dfrac{1}{8}\)

\(=x^3\cdot3\cdot\dfrac{1}{2}\cdot x^2+3\cdot\left(\dfrac{1}{2}\right)^2\cdot x+\left(\dfrac{1}{2}\right)^3\)

\(=\left(x+\dfrac{1}{2}\right)^3\)

e) \(27x^3-54x^2y+36xy^2-8y^3\)

\(=\left(3x\right)^3-3\cdot\left(3x\right)^2\cdot y+3\cdot x\cdot\left(2y\right)^2-\left(2y\right)^3\)

\(=\left(3x-2y\right)^3\)

a: =4x+2x^2-x^3-2x^2+x^3-4x+3=3

b: =24-4x+2x^2+3x^3-5x^2+4x+3x^2-3x^3=24

c: =x^4-x^3-3x^2+2x-x^4-x^3-3x^2+2x^2+2x+6+4x^2-4x-8

=-2

d: =x^2n-2x^n+x^n-2-x^2n+x^n+2009=2009