Bài 9: Phân tích đa thức thành nhân tử bằng cách phối hợp nhiều phương pháp

b: (x^2-1)^2-18(x+1)(x-1)

=(x+1)^2*(x-1)^2-18(x+1)(x-1)

=(x+1)(x-1)(x^2-1-18)

=(x+1)(x-1)(x^2-19)

c: (x+1)(x+3)(x+5)(x+7)+15

=(x^2+8x+7)(x^2+8x+15)+15

=(x^2+8x)^2+22(x^2+8x)+120

=(x^2+8x+10)(x^2+8x+12)
=(x^2+8x+10)(x+2)(x+6)

d: (x^2+x+4)^2+8x(x^2+x+4)+15x^2

=(x^2+x+4)^2+3x(x^2+x+4)+5x(x^2+x+4)+15x^2

=(x^2+x+4+3x)(x^2+x+4+5x)

=(x^2+6x+4)(x^2+4x+4)

=(x^2+6x+4)(x+2)^2

a: \(\dfrac{36}{x^6}-\dfrac{24}{x^3}+4\)

\(=4\left(\dfrac{9}{x^6}-\dfrac{6}{x^3}+1\right)\)
\(=4\left[\left(\dfrac{3}{x^3}\right)^2-2\cdot\dfrac{3}{x^3}\cdot1+1^2\right]\)

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

\(a,\dfrac{36}{x^6}-\dfrac{24}{x^3}+4\\ =4\left(\dfrac{9}{x^6}-\dfrac{6}{x^3}+1\right)\\ =4\left[\left(\dfrac{3}{x^3}\right)^2-2\cdot\dfrac{3}{x^3}\cdot1+1^2\right]\\ =4\left(\dfrac{3}{x^3}-1\right)^2\)

\(b,\left(x^2-1\right)^2-18\left(x+1\right)\left(x-1\right)\\ =\left(x^2-1\right)\left(x^2-1\right)-18\left(x^2-1\right)\\ =\left(x^2-1\right)\left(x^2-1-18\right)\\ =\left(x^2-1\right)\left(x^2-19\right)\\ =\left(x-1\right)\left(x+1\right)\left(x^2-19\right)\)

\(c,\left(x+1\right)\left(x+3\right)\left(x+5\right)\left(x+7\right)+15\\ =\left(x^2+8x+7\right)\left(x^2+8x+15\right)+15\\ =\left[\left(x^2+8x+11\right)-4\right]\left[\left(x^2+8x+11\right)+4\right]+15\\ =\left(x^2+8x+11\right)^2-16+15\\ =\left(x^2+8x+11\right)^2-1\\ =\left(x^2+8x+10\right)\left(x^2+8x+12\right)\)

\(d,\left(x^2+x+4\right)^2+8x\left(x^2+x+4\right)+15x^2\\ =\left(x^2+x+4\right)^2+3x\left(x^2+x+4\right)+5x\left(x^2+x+4\right)+15x^2\\ =\left(x^2+x+4\right)\left(x^2+4x+4\right)+5x\left(x^2+4x+4\right)\\ =\left(x^2+4x+4\right)\left(x^2+6x+4\right)\\ =\left(x+2\right)^2\left(x^2+6x+4\right)\)

64x^4+81

=64x^4+144x^2+81-144x^2

=(8x^2+9)^2-(12x)^2

=(8x^2-12x+9)(8x^2+12x+9)

x^8+4y^4

=x^8+4x^4y^2+4y^4-4x^4y^2

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

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

x^8+x^7+1

=x^8+x^7+x^6-x^6+1

=x^6(x^2+x+1)-(x^6-1)

=(x^2+x+1)*x^6-(x-1)(x+1)(x^2+x+1)(x^2-x+1)

=(x^2+x+1)[x^6-(x^2-1)(x^2-x+1)]

=(x^2+x+1)(x^6-x^4+x^2-x^2+x^2-x+1)

=(x^2+x+1)(x^6-x^4+x^2-x+1)

a) \(2x^2+5x+2\)

\(=2x^2+4x+x+2\)

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

\(=\left(x+2\right)\left(2x+1\right)\)

b) \(4x^2-4x-9y^2+12y-3\)

\(=\left(4x^2-4x+1\right)-\left(9y^2-12y+4\right)\)

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

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

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

c) \(x^4-2x^3-4x^2+4x-3\)

\(=x^4+x^3-x^2+x-3x^2-3x+3x-3\)

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

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

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

d) \(x^3-x+3x^2y+3xy^2+y^3-y\)

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

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

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

\(=\left(x+y\right)\left(x+y-1\right)\left(x+y+1\right)\)

\(a,a^2-2a-4b^2-4b\)

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

\(=\left(a-2b\right)\left(a+2b\right)-2\left(a+2b\right)\)

\(=\left(a+2b\right)\left(a-2b-2\right)\)

\(b,x^3-2x^2+4x-8\)

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

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

\(c,x^3+36x-12x^2\)

\(=x^3-6x^2-6x^2+36x\)

\(=x^2\left(x-6\right)-6x\left(x-6\right)\)

\(=\left(x-6\right)\left(x^2-6x\right)\)

\(=x\left(x-6\right)^2\)

\(d,5a^2+3\left(a+b\right)^2-5b^2\)

\(=\left(5a^2-5b^2\right)+3\left(a+b\right)^2\)

\(=5\left(a^2-b^2\right)+3\left(a+b\right)^2\)

\(=5\left(a-b\right)\left(a+b\right)+3\left(a+b\right)^2\)

\(=\left(a+b\right)\left[5\left(a-b\right)+3\left(a+b\right)\right]\)

\(=\left(a+b\right)\left(5a-5b+3a+3b\right)\)

\(=\left(a+b\right)\left(8a-2b\right)\)

\(=2\left(a+b\right)\left(4a-b\right)\)

\(e,x^3-3x^2+3x-1-y^3\)

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

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

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

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

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

#Urushi☕

\(c.\\ x^3+36x-12x^2\\ =x\left(x^2-12x+36\right)\\ =x.\left(x^2-2.x.6+6^2\right)\\ =x.\left(x-6\right)^2\\ ---\\ d.\\ 5a^2+3\left(a+b\right)^2-5b^2\\ =\left(5a^2-5b^2\right)+3\left(a+b\right)^2\\ =5.\left(a^2-b^2\right)+3.\left(a+b\right)\left(a+b\right)\\ =5\left(a+b\right)\left(a-b\right)+3\left(a+b\right)\left(a+b\right)\\ =\left(a+b\right)\left(5a-5b+3a+3b\right)\\ =\left(a+b\right)\left(8a-2b\right)\\ =2\left(a+b\right)\left(4a-b\right)\)

\(e.\\ x^3-3x^2+3x-1-y^3\\ =\left(x-1\right)^3-y^3\\ =\left(x-1-y\right)\left[\left(x-1\right)^2+\left(x-1\right).y+y^2\right]\\ =\left(x-y-1\right).\left[\left(x^2-2x+1\right)+y\left(x+y-1\right)\right]\)

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

\(b.\\ x^3-2x^2+4x-8=x^2\left(x-2\right)+4\left(x-2\right)\\ =\left(x^2+4\right)\left(x-2\right)\)

a: =(x^3-8y^3)-(x-2y)

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

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

b: =2(x^3-4x^2-12x+27)

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

=2(x+3)(x^2-7x+9)

e.

$x^4-4x^3-8x^2+8x=0$

$\Leftrightarrow x(x^3-4x^2-8x+8)=0$

$\Leftrightarrow x[x^2(x+2)-6x(x+2)+4(x+2)]=0$

$\Leftrightarrow x(x+2)(x^2-6x+4)=0$

$\Leftrightarrow x=0$ hoặc $x+2=0$ hoặc $x^2-6x+4=0$
$\Leftrightarrow x=0$ hoặc $x=-2$ hoặc $x^2-6x+4=0$

Xét riêng pt $x^2-6x+4=0$
$\Leftrightarrow (x-3)^2=5$

$\Leftrightarrow x-3=\pm \sqrt{5}$

$\Leftrightarrow x=3\pm \sqrt{5}$

Vậy.........

f.

$2x^2+3xy+y^2=0$

$\Leftrightarrow (2x^2+2xy)+(xy+y^2)=0$

$\Leftrightarrow 2x(x+y)+y(x+y)=)$

$\Leftrightarrow (x+y)(2x+y)=0$

$\Leftrightarrow x+y=0$ hoặc $2x+y=0$

$\Leftrightarrow x=-y$ hoặc $x=\frac{-y}{2}$

g.

$2x^4-x^3-9x^2+13x-5=0$

$\Leftrightarrow (2x^4-2x^3)+(x^3-x^2)-(8x^2-8x)+(5x-5)=0$

$\Leftrightarrow 2x^3(x-1)+x^2(x-1)-8x(x-1)+5(x-1)=0$

$\Leftrightarrow (x-1)(2x^3+x^2-8x+5)=0$

$\Leftrightarrow (x-1)[2x^2(x-1)+3x(x-1)-5(x-1)]=0$

$\Leftrightarrow (x-1)^2(2x^2+3x-5)=0$

$\Leftrightarrow (x-1)^2[2x(x-1)+5(x-1)]=0$

$\Leftrightarrow (x-1)^3(2x+5)=0$

$\Leftrightarrow x-1=0$ hoặc $2x+5=0$

$\Leftrightarrow x=1$ hoặc $x=\frac{-5}{2}$

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

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

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

b: =3(3x^2-2x-1)

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

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

c: =2x^2-4x+x-2

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

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

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

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

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

e: =3x^2+9x+x+3

=3x(x+3)+(x+3)

=(x+3)(3x+1)

a) \(4x^2+5x-6\)

\(=4x^2+8x-3x-6\)

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

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

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

b) \(9x^2-6x-3\)

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

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

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

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

c) \(2x^2-3x-2\)

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

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

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

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

d) \(3x^2+x-2\)

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

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

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

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

e) \(3x^2+10x+3\)

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

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

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

1.

= (x^3 + 125 ) -(x^2 +5x)

=(x +5) (x^2 -5x +25) -x(x+5)

=(x+5)(x^2 -5x +25 -x)

=(x+5)(x^2 -6x +25)

2.

= (x^3 -27) + (2x^2 -6x)

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

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

=(x-3) (x^2 +5x +9)

2x - 2y - x² + 2xy - y²

= (2x - 2y) - (x² - 2xy + y²)

= 2(x - y) - (x - y)²

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

\(a,A=y^2-\dfrac{1}{2}y+\dfrac{1}{16}\)

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

\(=\left(y-\dfrac{1}{4}\right)^2\)

Với \(y=100,25\), ta được:

\(A=\left(100,25-\dfrac{1}{4}\right)^2\)

\(=\left(\dfrac{401}{4}-\dfrac{1}{4}\right)^2\)

\(=\left(\dfrac{400}{4}\right)^2=100^2=10000\)

\(------\)

\(b,B=4x^2-9y^2-6y-1\)

\(=\left(2x\right)^2-\left[\left(3y\right)^2+2.3y.1+1\right]\)

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

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

Với \(x=23;y=1\), ta được:

\(B=\left(2.23-3.1-1\right)\left(2.23+3.1+1\right)\)

\(=\left(46-4\right)\left(46+4\right)\)

\(=42.50=2100\)