a) \(27x^3+27x^2+9x+1\\ =\left(3x\right)^3+3.\left(3x\right)^2.1+3.3x.1^2+1^3\\ =\left(3x+1\right)^3\)

Thay x=13 vào biểu thức, ta có:

\(\left(3x+1\right)^3=\left(3.13+1\right)^3=40^3=64000\)

b) \(x^3+12x^2+48x+65\\ =\left(x^3+3.x^2.4+3.x.4^2+4^3\right)+1\\ =\left(x+4\right)^3+1\)

Thay x=6 vào biểu thức, ta có:

\(\left(x+4\right)^3+1=\left(6+4\right)^3+1=10^3+1=1000+1=1001\)

a) \(27x^3+27x^2+9x+1\)

= \(\left(3x\right)^3+3.\left(3x\right)^2.1+3.1^2.3x+1^3\)

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

Thay x=13 vào pt(1),ta được:

\(\left(3x+1\right)^3=\left(3.13+1\right)^3=40^3=64000\)

b) \(x^3+12x^2+48x+65\)

= \(x^3+3.x^2.4+3.4^2.x+4^3+1\)

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

Thay x=6 vào pt(2),ta được:

\(\left(x+4\right)^3+1=\left(6+4\right)^3+1=10^3+1=1001\)

b) \(x^3-15x^2+75x-125\)

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

Thay x = 35 vào ta đc:

\(\left(35-5\right)^3\) = 27000

c) \(x^3+12x^2+48x+65\)

= \(x^3+5x^2+7x^2+13x+65\)

= \(x^2\left(x+5\right)+7x\left(x+5\right)+13\left(x+5\right)\)

= \(\left(x+5\right)\left(x^2+7x+13\right)\)

Thay x = 6 vào ta đc:

\(\left(6+5\right)\left(6^2+7.6+13\right)\)=1001

2)

Ta có: a+b+c=0

=> \(\left(a+b+c\right)^3=0\)

=> \(a^3+b^3+c^3+3a^2b+3ab^2+3b^2c+3bc^2+3a^2c+3ac^2+6abc=0\)=> \(a^3+b^3+c^3+\left(3a^2b+3ab^2+3abc\right)+\left(3b^2c+3bc^2+3abc\right)+\left(3a^2c+3ac^2+3abc\right)-3abc=0\)

=> \(a^3+b^3+c^3+3b\left(a+b+c\right)+3bc\left(a+b+c\right)+3ac\left(a+b+c\right)=3abc\)

Do a+b+C = 0

=> \(a^3+b^3+c^3=3abc\) (đpcm)

a ) \(27x^3+27x^2+9x+1\)

\(=\left(3x\right)^3+3\left(3x\right)^2+3.3x+1\)

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

Thay \(x=13\) vào b/t trên ta được :

\(\left(3.13+1\right)^3=40^3=64000\)

Vậy g/t b/t trên là : \(64000\) tại \(x=13\)

b ) \(x^3-15x^2+75x-125\)

\(=x^3-3x^2.5+3x.5^2-5^3\)

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

Thay \(x=35\) vào b/t trên ta được :

\(\left(35-5\right)^3=30^3=27000\)

Vậy g/t b/t trên là : \(27000\Leftrightarrow x=35\)

c ) \(x^3+12x^2+48x+65\)

\(=x^3+3x^2.4+3x.4^2+4^3+1\)

\(=\left(x+4\right)^3+1\)

Thay \(x=6\) vào b/t trên , ta được :

\(\left(6+4\right)^3+1=10^3+1=1000+1=1001\)

Vậy g/t b/t trên là : \(1001\) tại \(x=6\)

a) \(27x^3+27x^2+9x+1\)

\(=\left(3x\right)^3+3.\left(3x\right)^2+3.3x+1^3\)

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

Thay x = 13, ta được:

\(=\left(3.13+1\right)^3\)

\(=40^3\)

\(=64000\)

b) \(x^3-15x^2+75x-125\)

\(=x^3-3.x^2.5+3.x.5^2-5^3\)

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

Thay x = 35, ta được:

\(=\left(35-5\right)^3\)

\(=30^3\)

\(=27000\)

c) \(x^3+12x^2+48x+65\)

\(=x^3+5x^2+7x^2+35x+13x+65\)

\(=x^2\left(x+5\right)+7x\left(x+5\right)+13\left(x+5\right)\)

\(=\left(x+5\right)\left(x^2+7x+13\right)\)

Thay x = 6, ta được:

\(=\left(6+5\right)\left(6^2+7.6+13\right)\)

\(=1001\)

a) x³ - 9x² + 27x - 27 = -8

<=> x³ - 3x².3 + 3x.3² - 3³ = -8

<=> (x - 3)³ = -2³

<=> x - 3 = -2

<=> x = 1 (T/m)

Vậy x = 1.

b) 64x³ + 48x² + 12x + 1 = 27

<=> (4x)³ + 3.(4x)².1 + 3.4x.1² + 1³ = 27

<=> (4x + 1)³ = 3³

<=> 4x + 1 = 3

<=> 4x = 2

<=> x = \(\frac{1}{2}\)(T/m)

Vậy x = \(\frac{1}{2}\).

c) (2x - 1)³ - 4x².(2x - 3) = 5

<=> (8x³ - 12x² + 6x - 1) - (8x³ - 12x²) = 5

<=> 8x³ - 12x² + 6x - 1 - 8x³ + 12x² = 5

<=> 6x - 1 = 5

<=> 6x = 6

<=> x = 1 (T/m)

Vậy x = 1.

\(x^3-9x^2+27x-27=-8\Leftrightarrow\left(x^3-27\right)-\left(9x^2-27x\right)=\left(x-3\right)\left(x^2+3x+9\right)-9x\left(x-3\right)=\left(x-3\right)\left(x^2-6x+9\right)=\left(x-3\right)^3=-8=\left(-2\right)^3\Rightarrow x=\left(-2\right)+3=1\)

\(64x^3+48x^2+12x+1=\left(64x^3+1\right)+\left(48x^2+12x\right)=\left(4x+1\right)\left(16x^2-4x+1\right)+12x\left(4x+1\right)=\left(4x+1\right)\left(16x^2+8x+1\right)=\left(4x+1\right)^3=27\Rightarrow4x=2\Leftrightarrow x=\frac{1}{2}\)

c) \(\left(2x-1\right)^3-4x^2.\left(2x-3\right)=5\)

\(\Leftrightarrow\left(8x^3-12x^2+6x-1\right)-\left(8x^3-12x^2\right)=5\)

\(\Leftrightarrow8x^3-12x^2+6x-1-8x^3+12x^2=5\)

\(\Leftrightarrow6x-1=5\)

\(\Leftrightarrow6x=6\)

\(\Leftrightarrow x=1\)

d) \(\left(x+4\right)^3-x^2.\left(x+12\right)=16\)

\(\Leftrightarrow\left(x^3+12x^2+48x+64\right)-\left(x^3+12x^2\right)=16\)

\(\Leftrightarrow x^3+12x^2+48x+64-x^3-12x^2=16\)

\(\Leftrightarrow48x+64=16\)

\(\Leftrightarrow48x=-48\)

\(\Leftrightarrow x=-1\)

#vì câu a,b có người làm rồi nên mình chỉ làm c,d thôi nhé ! :)

Học Tốt !!

a: \(=\left(x+1+5\right)\left(x+1-5\right)=\left(x+6\right)\left(x-4\right)\)

b: =(1-2x)(1+2x)

c: \(=\left(2-3x\right)\left(4+6x+9x^2\right)\)

d: =(x+3)^3

e: \(=\left(2x-y\right)^3\)

f: =(x+2y)(x^2-2xy+4y^2)