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1, \(25x^2-10xy+y^2=\left(5x-y\right)^2\)
2, \(8x^3+36x^2y+54xy^2+27y^3=\left(2x+3y\right)^3\)
4, \(\left(a+b+c\right)^3-a^3-b^3-c^3\)
\(=a^3+b^3+c^3+3\left(a+b\right)\left(b+c\right)\left(a+c\right)-a^3-b^3-c^3\)
\(=3\left(a+b\right)\left(b+c\right)\left(a+c\right)\)
5, \(2x^3+3x^2+2x+3\)
\(=x^2\left(2x+3\right)+2x+3\)
\(=\left(x^2+1\right)\left(2x+3\right)\)
6, \(x^3z+x^2yz-x^2z^2-xyz^2\)
\(=x^3z-x^2z^2+x^2yz-xy^2\)
\(=xz\left(x^2-xz\right)+xz\left(xy-yz\right)\)
\(=xz\left[x\left(x-z\right)+y\left(x-z\right)\right]\)
\(=xz\left(x+y\right)\left(x-z\right)\)
8, \(x^3+3x^2y+3xy^2+y+y^3\)\(=\left(x+y\right)^3+y\)
9, \(x^2-6x+8\)
\(=x^2-4x-2x+8\)
\(=x\left(x-4\right)-2\left(x-4\right)\)
\(=\left(x-2\right)\left(x-4\right)\)
10, \(x^2-8x+12\)
\(=x^2-6x-2x+12\)
\(=x\left(x-6\right)-2\left(x-6\right)\)
\(=\left(x-2\right)\left(x-6\right)\)
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11, \(a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left(a-b\right)\)
\(=a^2b-a^2c+b^2c-b^2a+c^2a-c^2b\)
\(=a^2b-ab^2+abc-a^2c+b^2c-abc+ac^2-c^2b\)
\(=ab\left(a-b\right)-ac\left(a-b\right)-bc\left(a-b\right)+c^2\left(a-b\right)\)
\(=\left(a-b\right)\left(ab-ac-bc+c^2\right)\)
\(=\left(a-b\right)\left[b\left(a-c\right)-c\left(a-c\right)\right]\)
\(=\left(a-b\right)\left(a-c\right)\left(b-c\right)\)
12, \(x^3-7x-6\)
\(=x^3-3x^2+3x^2-9x+2x-6\)
\(=x^2\left(x-3\right)+3x\left(x-3\right)+2\left(x-3\right)\)
\(=\left(x-3\right)\left(x^2+3x+2\right)\)
\(=\left(x-3\right)\left(x^2+x+2x+2\right)\)
\(=\left(x-3\right)\left[x\left(x+1\right)+2\left(x+1\right)\right]\)
\(=\left(x-3\right)\left(x+2\right)\left(x+1\right)\)
13, \(x^4+4\)
\(=x^4+4x^2+4-4x^2\)
\(=\left(x^2+2\right)^2-4x^2\)
\(=\left(x^2-2x+2\right)\left(x^2+2x+2\right)\)
14, \(a^4+64\)
\(=a^4+16a^2+64-16a^2\)
\(=\left(a^2+8\right)^2-16a^2\)
\(=\left(a^2-4a+8\right)\left(a^2+4a+8\right)\)
15, \(x^5+x+1\)
\(=x^5-x^2+x^2+x+1\)
\(=x^2\left(x^3-1\right)+x^2+x+1\)
\(=x^2\left(x-1\right)\left(x^2+x+1\right)+x^2+x+1\)
\(=\left(x^2+x+1\right)\left[x^2\left(x-1\right)+1\right]\)
16, \(x^5+x-1\)
\(=x^5-x^4+x^3+x^4-x^3+x^2-x^2+x-1\)
\(=x^3\left(x^2-x+1\right)-x^2\left(x^2-x+1\right)-\left(x^2-x+1\right)\)
\(=\left(x^2-x+1\right)\left(x^3-x^2-1\right)\)
17, \(\left(x^2+x\right)^2-2\left(x^2+x\right)-15\)
\(=\left(x^2+x\right)\left(x^2+x-2\right)-15\)
19, \(\left(x^2+8x+7\right)\left(x^2+8x+15\right)+15\) (*)
Đặt \(x^2+8x+7=a\) ta có:
(*) \(\Leftrightarrow a\left(a+8\right)+15\)
\(\Leftrightarrow a^2+8a+15\)
\(\Leftrightarrow a^2+3a+5a+15\)
\(\Leftrightarrow a\left(a+3\right)+5\left(a+3\right)\)
\(\Leftrightarrow\left(a+3\right)\left(a+5\right)\)
Trả lại biến cũ ta có: (*) \(\Leftrightarrow\left(x^2+8x+10\right)\left(x^2+8x+12\right)\)
20, \(\left(x^2+3x+1\right)\left(x^2+3x+2\right)-6\) (*)
Đặt \(x^2+3x+1=a\) ta có:
(*) \(\Leftrightarrow a\left(a+1\right)-6\)
\(\Leftrightarrow a^2+a-6\)
\(\Leftrightarrow a^2+3a-2a-6\)
\(\Leftrightarrow a\left(a+3\right)-2\left(a+3\right)\)
\(\Leftrightarrow\left(a-2\right)\left(a+3\right)\)
Trả lại biến cũ ta có: (*) \(\Leftrightarrow\left(x^2+3x-1\right)\left(x^2+3x+5\right)\)
Bài 1:
a) 25x2 - 10xy + y2 = (5x - y)2
b) 81x2 - 64y2 = (9x)2 - (8y)2 = (9x - 8y)(9x + 8y)
c) 8x3 + 36x2y + 54xy2 + 27y3
= 8x3 + 27y3 + 36x2y + 54xy2
= (2x + 3y)(4x2 - 6xy + 9y2) + 18xy(2x + 3y)
= (2x + 3y)(4x2 - 6xy + 18xy + 9y2)
= (2x + 3y)(4x2 + 12xy + 9y2)
= (2x + 3y)(2x + 3y)2 = (2x + 3y)3
c) (a2 + b2 - 5)2 - 4(ab + 2)2 = (a2 + b2 - 5)2 - 22(ab + 2)2
= (a2 + b2 - 5)2 - (2ab + 4)2
= (a2 + b2 - 5 - 2ab - 4)(a2 + b2 - 5 + 2ab + 4)
= (a2 - 2ab + b2 - 9)(a2 + 2ab + b2 - 1)
= \(\left [ (a - b)^{2} - 3^{2} \right ]\)\(\left [ (a + b)^{2} - 1\right ]\)
= (a - b - 3)(a - b + 3)(a + b - 1)(a + b + 1)
pn đăng mỗi lần vài bài thôi chứ đăng nhìn ngán lắm
Bài 2:
a) 2x3 + 3x2 + 2x + 3
= 2x3 + 2x + 3x2 + 3
= 2x(x2 + 1) + 3(x2 + 1)
= (x2 + 1)(2x + 3)
b)x3z + x2yz - x2z2 - xyz2
= xz(x2 + xy - xz - yz)
= \(xz\left [ x(x + y) - z(x + y) \right ]\)
= xz(x + y)(x - z)
c) x2y + xy2 - x - y
= xy(x + y) - (x + y)
= (x + y)(xy - 1)
d) 8xy3 - 5xyz - 24y2 + 15z
= 8xy3 - 24y2 - 5xyz + 15z
= 8y2(xy - 3) - 5z(xy - 3)
= (xy - 3)(8y2 - 5z)
e) x3 + y(1 - 3x2) + x(3y2 - 1) - y3
= x3 - y3 + y - 3x2y + 3xy2 - x
= (x - y)(x2 + xy + y2) - 3xy(x - y) - (x - y)
= (x - y)(x2 + xy + y2 - 3xy - 1)
= (x - y)(x2 - 2xy + y2 - 1)
= \((x - y)\left [ (x - y)^{2} - 1 \right ]\)
= (x - y)(x - y - 1)(x - y + 1)
câu f tương tự
2x^4+x^3-5x^2-3x-3 2x^2+x+1 x^2-3 2x^4 -6x^2 - x^3+x^2-3x-3 x^3 -3x - x^2-3 x^2-3 x^2-3 - 0
x^3-3x^2+x-3 x-3 x^2+1 x^3-3x^2 - - x+3 x+3 - 0