Bài 1:
a.
\(27(1-x)(x^2+x+1)+81x(x-1)\)
\(=27(1-x^3)+81x^2-81x\)
\(=3^3-3.3^2.3x+3.3.(3x)^2-(3x)^3\)
\(=(3-3x)^3=27(1-x)^3\)
b.
\(y[x^2+x(x-y)+(x-y)^2]+(x-y)^3\)
\(=[x-(x-y)][x^2+x(x-y)+(x-y)^2]+(x-y)^3\)
\(=x^3-(x-y)^3+(x-y)^3=x^3\)
Bài 12:
Ta có: a+b+c=0
nên \(\left\{{}\begin{matrix}a+b=-c\\a+c=-b\\b+c=-a\end{matrix}\right.\)
Ta có: \(a+b+c=0\)
\(\Leftrightarrow\left(a+b+c\right)^3=0\)
\(\Leftrightarrow a^3+b^3+c^3+3\left(a+b\right)\left(a+c\right)\left(b+c\right)=0\)
\(\Leftrightarrow a^3+b^3+c^3=3abc\)
Bài 2:
a.
\(A=3(x^2-2x+1)-(x^2+2x+1)+2(x^2-9)-(4x^2+12x+9)+20x-5\)
\(=3x^2-6x+3-x^2-2x-1+2x^2-18-4x^2-12x-9+20x-5\)
\(=(3x^2-x^2+2x^2-4x^2)+(-6x-2x-12x+20x)+(3-1-18-9-5)\)
\(=-30\) không phụ thuộc vào giá trị của $x$
b.
\(B=-x(x^2+4x+4)+(4x^2+4x+1)+x^3+27-1=27\)
Vậy $B$ không phụ thuộc vào giá trị $x$
Bài 3:
a.
\(A=2(x+y)(x^2-xy+y^2)-3(x^2+y^2)\)
\(=2(x^2-xy+y^2)-3(x^2+y^2)=-2xy-x^2-y^2=-(x^2+2xy+y^2)\)
\(=-(x+y)^2=-1^2=-1\)
b.
\(B=x^3+y^3+3xy(x+y)=(x+y)^3=1^3=1\)
c.
\(C=(2x-3y)(4x^2+6xy+9y^2)\)
\(=4x^2+6xy+9y^2=(2x-3y)^2+18xy\)
\(=5^2+18.4=97\)
Bài 4:
Theo HĐT đáng nhớ:
$(a+b)^3=a^3+3a^2b+3ab^2+b^3=a^3+b^3+3ab(a+b)$. Áp dụng vào bài:
\((a+b+c)^3=(a+b)^3+c^3+3(a+b)c(a+b+c)\)
\(=a^3+b^3+3ab(a+b)+c^3+3(a+b)(c^2+ac+bc)\)
\(=a^3+b^3+c^3+3(a+b)(ab+c^2+ac+bc)\)
\(=a^3+b^3+c^3+3(a+b)[a(b+c)+c(c+b)]\)
\(=a^3+b^3+c^3+3(a+b)(b+c)(a+c)\)
Bài 5:
Vì $a+b+c=0$ nên $a+b=-c; b+c=-a; c+a=-b$
Khi đó, áp dụng bài 4 ta có:
$(a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(c+a)$
$\Leftrightarrow 0^3=a^3+b^3+c^3+3(-c)(-a)(-b)$
$\Leftrightarrow 0=a^3+b^3+c^3-3abc$
$\Leftrightarrow a^3+b^3+c^3=3abc$ (đpcm)
Bài 10:
a: Ta có: \(A=2\left(x^3+y^3\right)-3\left(x^2+y^2\right)\)
\(=2\left(x^2-xy+y^2\right)-3\cdot\left(-2xy\right)\)
\(=2x^2+2xy+2y^2+6xy\)
\(=2x^2+8xy+2y^2\)
b: Ta có: \(B=x^3+y^3+3xy\)
\(=x^2-xy+y^2+3xy\)
\(=x^2+2xy+y^2\)
c: Ta có: \(C=8x^3-27y^3\)
\(=\left(2x-3y\right)^3+3\cdot2x\cdot3y\cdot\left(2x-3y\right)\)
\(=5^3+3\cdot6\cdot4\cdot5\)
\(=485\)
Bài 8:
a: Ta có: \(27\left(1-x\right)\left(x^2+x+1\right)+81x\left(x-1\right)\)
\(=27\left(1-x^3\right)+81x^2-81x\)
\(=27-27x^3+81x^2-81x\)
b: Ta có: \(y\left[x^2+x\left(x-y\right)+\left(x-y\right)^2\right]+\left(x-y\right)^3\)
\(=y\left(x^2+x^2-xy+x^2-2xy+y^2\right)+\left(x-y\right)^3\)
\(=y\left(3x^2-3xy+y^2\right)+\left(x-y\right)^3\)
\(=3x^2y-3xy^2+y^3+x^3-3x^2y+3xy^2-y^3\)
\(=x^3\)
Bài 9:
a: Ta có: \(A=3\left(x-1\right)^2-\left(x+1\right)^2+2\left(x-3\right)\left(x+3\right)-\left(2x+3\right)^2-\left(5-20x\right)\)
\(=3x^2-6x+3-x^2-2x-1+2x^2-18-4x^2-12x-9-5+20x\)
=-16
b: Ta có: \(B=-x\left(x+2\right)^2+\left(2x+1\right)^2+\left(x+3\right)\left(x^2-3x+9\right)-1\)
\(=-x^3-4x^2-4x+4x^2+4x+1+x^3-27-1\)
=-27
