\(\left(a+b\right)^3-3ab\left(a+b\right)\)

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

\(=a^3+b^3\)

Lời giải:

$(a+b)^3-3ab(a+b)$

$=a^3+3a^2b+3ab^2+b^3-(3a^2b+3ab^2)$

$=a^3+b^3$
Ta có đpcm.

\(VP=\left(a+b\right)^3-3ab\left(a+b\right)=a^3+3a^2b+3ab^2+b^3-3a^2b-3ab^2=a^3+b^3=VT\)

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

\(\left(a+b\right)^3-3ab\left(a+b\right)\)

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

\(=a^3+b^3\)

\(a,=\left(x+3\right)^3=\left(-3+3\right)^3=0\\ b,=27x^3+1-\left(1-27x^3\right)=27x^3+1-1+27x^3=54x^3\\ =54\cdot10^3=54\cdot1000=54000\)

c, hình như sai đề á e

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

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

\(=\left(5x\right)^3+\left(3y\right)^3=125x^3-27y^3\)

a) \(\left(a+b\right)^3-\left(a-b\right)^3=a^3+3a^2b+3ab^2+b^3-\left(a^3-3a^2b+3ab^2-b^3\right)=6a^2b+b^3=b\left(6a^2+b^2\right)\)

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

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

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

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

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

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

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

a: \(\left(a+b\right)^3-\left(a-b\right)^3\)

\(=a^3+3a^2b+3ab^2+b^3-a^3+3a^2b-3a^2b+b^3\)

\(=6a^2b+2b^3\)

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

Biến đổi VP

=> VT = VP

=> Đpcm

\(a,\Leftrightarrow\left(x^2-3\right)^2=0\\ \Leftrightarrow x^2-3=0\\ \Leftrightarrow\left(x-\sqrt{3}\right)\left(x+\sqrt{3}\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=\sqrt{3}\\x=-\sqrt{3}\end{matrix}\right.\\ b,\Leftrightarrow8x^3+12x^2+6x+1-64=0\\ \Leftrightarrow\left(2x+1\right)^3-4^3=0\\ \Leftrightarrow\left(2x+1-4\right)\left[\left(2x+1\right)^2+4\left(2x+1\right)+16\right]=0\\ \Leftrightarrow\left[{}\begin{matrix}2x=3\\4x^2+4x+1+8x+4+16=0\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=\dfrac{3}{2}\\4x^2+12x+17=0\left(1\right)\end{matrix}\right.\)

Xét \(\left(1\right)\Leftrightarrow\left(2x+3\right)^2+8=0\left(vô.lí\right)\Leftrightarrow x\in\varnothing\)

Vậy pt có nghiệm \(x=\dfrac{3}{2}\)

\(c,\Leftrightarrow\left(3-2x-5\right)\left(3-2x+5\right)=0\\ \Leftrightarrow\left(-2-2x\right)\left(8-2x\right)=0\\ \Leftrightarrow-2\left(x+1\right)\cdot2\left(4-x\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=-1\\x=4\end{matrix}\right.\) 

a)
\(\left(3x\right)^3-3.\left(3x\right)^2.1+3.3x.2^2-2^3=0\)
\(\left(3x-2\right)^3=0\)
3x-2=0
3x=2
x=2/3
b)
\(x^3-3.x^2.5+3.x.5^2+5^3=0\)
\(\left(x-5\right)^3=0\)
x-5=0
x=5