Bài 1 : Phân tích các đa thức sau thành nhân tử :

a) 8x³ - 64

=(2x)³ + 4³

=(2x+4)(4x² - 8x + 16)

c) 125x³ + 1

=5x³ + 1³

=(5x+1)(25x² +5x+1)

d) 8x³ - 27

=(2x)³ - 3³

=(2x - 3)(2x² + 6x + 9)

e) 1 + 8x⁶y³

=1 + (2x²y)³

=(1 + 2x²y)(4x⁴y² -2x²y + 1)

f) 125x³ + 27y³

=(5x)³ + (3y³)

=(5x + 3y)(25x² - 15xy + 9y²)

Bài 1

a) \(8x^3-64\)

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

\(=\left(2x-4\right)\left(4x^2+8x+16\right)\)

c) \(125x^3+1\)

\(=\left(5x\right)^3+1^3\)

\(=\left(5x+1\right)\left(25x^2-5x+1\right)\)
d) \(8x^3-27\)

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

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

e) \(1+8x^6x^3\)

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

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

f) \(125x^3+27y^3\)

\(=\left(5x\right)^3+\left(3y\right)^3\)

\(=\left(5x+3y\right)\left(25x^2-15xy+9x^2\right)\)

Bài 2

b) \(x^3+6x^2+12x+8\)

\(=x^3+3.x^2.2+3.x.2^2+2^3\)

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

c) \(x^3-3x^2+3x-1\)

\(=x^3-3.x^2.1+3.x.1^2-1^3\)

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

d) \(27x^3-54x^2y+36xy^2-8y^3\)

\(=\left(3x\right)^3-3.\left(3x\right)^2.2y+3.3x.\left(2y\right)^2-\left(2y\right)^3\)

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

a) 8x³ - 64 = (2x)³ - 4³

= (2x - 4)\([\)(2x)² + 2x.4 + 4²\(]\)

= (2x - 4)(4x² + 8x + 16)

b) 1 + 8x⁶y³

= 1³ + (2x²y)³

= (1 + 2x²y)[(2x²y)² - 2x²y.1 + 1²]

= (1 + 2x²y)(4x⁴y² - 2x²y + 1)

c) 27x³ + \(\frac{y^3}{8}\)

= (3x)³ + \(\left(\frac{y}{2}\right)^3\)

= \(\left(3x+\frac{y}{2}\right)\left[\left(3x\right)^2-3x.\frac{y}{2}+\left(\frac{y}{2}\right)^2\right]\)

= \(\left(3x-\frac{y}{2}\right)\left(9x^2-\frac{3xy}{2}+\frac{y^2}{4}\right)\)

d) 125x³ + 27y³

= (5x)³ + (3y)³

= (5x + 3y)[(5x)² - 5x.3y + (3y)²]

= (5x + 3y)(25x² - 15xy + 9y²)

a) \(8x^3-64\)

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

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

\(=\left(2x-4\right)\left(4x^2+8x+16\right)\)

d) \(125x^3+27y^3\)

\(=\left(5x\right)^3+\left(3y\right)^3\)

\(=\left(5x+3y\right)\left[\left(5x\right)^2-5x.3y+\left(3y\right)^2\right]\)

\(=\left(5x+3y\right)\left(25x^2-15xy+9y^2\right)\)

Câu b có sai đề không ạ ?

\(\left(x-1\right)^2-25\)

\(=x^2-2x+1-25\)

\(=x^2-2x-24\)

\(=x^2-6x+4x-24\)

\(=x.\left(x-6\right)+4.\left(x-6\right)\)

\(=\left(x+4\right).\left(x-6\right)\)

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

b, \(\left(x+1\right)^2-25=\left(x+1\right)^2-5^2=\left(x+1-5\right)\left(x+1+5\right)=\left(x-4\right)\left(x+6\right)\)

c, \(1-4x^2=1^2-\left(2x\right)^2=\left(1-2x\right)\left(1+2x\right)\)

d,  \(8-27x^3=2^3-\left(3x\right)^3=\left(2-3x\right)\left(4+6x+9x^2\right)\)

a)=(1-y)²

b)=(x+1)²-5²

=(x+1+5)(x+1-5)

=(x+6)(x-4)

c)=1²-(2x)²

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

d)=2³-(3x)³

=(2-3x)(4+6x+9x²)

e)=3³+3.9.x+3.3.x²+x³

=(3+x)³

1) \(\dfrac{1}{27}+a^3=\left(\dfrac{1}{3}+a\right)\left(\dfrac{1}{9}-\dfrac{a}{3}+a^2\right)\)

2) \(=\left(2x+3y\right)\left(4x^2-6xy+9y^2\right)\)

3) \(=\left(\dfrac{1}{2}x+2y\right)\left(\dfrac{1}{4}x-xy+4y^2\right)\)

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

5) \(=\left(x^3+1\right)\left(x^6-x^3+1\right)\)

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

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

8) \(=\left(2x^2-3y\right)\left(4x^4+6x^2y+9y^2\right)\)

9) \(=\left(\dfrac{1}{4}x^2-5y\right)\left(\dfrac{1}{16}x^4+\dfrac{5}{4}x^2y+25y^2\right)\)

10) \(=\left(\dfrac{1}{2}x-2\right)\left(\dfrac{1}{4}x^2+x+4\right)\)

11) \(=\left(x+2\right)^3\)

12) \(=\left(x+3\right)^3\)

Bài làm:

a, 1-4x²

=1-(2x)²

=(1-2x).(1+2x)

b, 8-27x³

=2³-(3x)³

=(2-3x).(4+6x+9x²)

Các câu còn lại bạn dùng hằng đẳng thức là phân tích được ra thôi

1 - 4x^2 

= 1^2 - ( 2x )^2 

= ( 1 - 2x ) ( 1 + 2x ) 

8 - 27x^ 3 

= 2^3 - ( 3x )^3 

= ( 2 - 3x ) [ 2^2 + 2 * 3x + ( 3x )^2 ]

= ( 2 - 3x ) ( 4 + 6x + 9x^2 ) 

= ( 2 - 3x ) ( 9x^2 + 6x + 4 ) 

27 + 27x + 9x^2 + x^3 

= x^3 + 9x^2 + 27x + 27 

= x^3 + 3x^2 + 6x^2 + 18x + 9x + 27 

= x^2 ( x + 3 ) + 6x ( x + 3 ) + 9 ( x + 3 ) 

= ( x + 3 ) ( x^2 + 6x + 9 ) 

= ( x + 3 ) ( x + 3 )^2 

= ( x + 3 )^3 

x^2 + 4x - 5 

= x^2 - x + 5x - 5 

= x ( x - 1 ) + 5 ( x - 1 ) 

= ( x + 1 ) ( x - 5 ) 

a) 8x³ - 125=(2x)³-5³=(2x-5)(2x+5)

b) 1 - 27x³y⁶=1-3³x³(y²)³=1-(3xy²)³=(1-3xy²)(1-3xy²+9x²y⁴)

c)27x³ + y³/64=(3x)³ + y³/4³=(3x)³ + (y/4)³=(3x+y/4)(9x²-3xy/4+y²/16)

d) 27x³ + 125y³ = (3x)³+(5y)³=(3x+5y)(9x²-15xy+25y²)