Câu 1:

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

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

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

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

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

Câu 2:

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

\(=a\left(b^3-c^3\right)+bc^3-a^3b+a^3c-b^3c\)

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

\(=\left(b-c\right)\left(ab^2+abc+c^2a-a^3-b^2c-bc^2\right)\)

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

\(=\left(b-c\right)\left(c-a\right)\left(ca+a^2-b^2-bc\right)\)

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

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

\(ĐK:-3\le x\le6\)

Đặt \(t=\sqrt{x+3}+\sqrt{6-x}\left(t>0\right)\Rightarrow t^2=9+2\sqrt{\left(x+3\right)\left(6-x\right)}\Rightarrow\sqrt{\left(x+3\right)\left(6-x\right)}=\frac{t^2-9}{2}\)

Phương trình trở thành \(t-\frac{t^2-9}{2}=3\Leftrightarrow t^2-2t-3=0\Leftrightarrow\left(t-3\right)\left(t+1\right)=0\Leftrightarrow\orbr{\begin{cases}t=3\left(tm\right)\\t=-1\left(L\right)\end{cases}}\)    

Với t = 3 thì \(\sqrt{x+3}+\sqrt{6-x}=3\Rightarrow\sqrt{\left(x+3\right)\left(6-x\right)}=0\Rightarrow\orbr{\begin{cases}x=6\\x=-3\end{cases}}\left(tm\right)\)  

Vậy phương trình có tập nghiệm S = {6; -3}

Câu 1:

\(a^2+2ab+b^2-ac-bc\)

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

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

Câu 2:

\(5x^2-5y^2-10x+10y\)

\(=5\left(x-y\right)\left(x+y\right)-10\left(x-y\right)\)

\(=\left(x-y\right)\left(5x+5y-10\right)\)

\(=5\left(x-y\right)\left(x+y-2\right)\)

Câu 3:

\(3x^2-6xy+3y^2-12z^2\)

\(=3\left[\left(x-y\right)^2-4z^2\right]\)

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

Câu 4:

\(x^4+x^3+x^2-1\)

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

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

Câu 5:

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

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

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

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

Câu 6:

\(x^4-x^2+2x-1\)

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

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

Câu 7:

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

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

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

a)

\(=\frac{\left(\sqrt{a}+1\right)^2-\left(\sqrt{a}-1\right)^2+4\sqrt{a}\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}.\frac{1}{2a\sqrt{a}}\)

\(=\frac{a+2\sqrt{a}+1-a+2\sqrt{a}-1+4a\sqrt{a}-4\sqrt{a}}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}.\frac{1}{2a\sqrt{a}}\)

\(=\frac{4a\sqrt{a}}{a-1}.\frac{1}{2a\sqrt{a}}=\frac{2}{a-1}\)

b) \(\frac{2}{a-1}=a\Rightarrow a^2-a-2=0\)

Ta có: 1+1+(-2)=0, nên pt có 2 nghiệm a₁=-1<0 (không thỏa mãn đk)=> loại

a₂=2(thỏa mãn đk)=> chọn

Vậy a=2 thì P=a

\(A=\frac{\sqrt{x}+1}{\sqrt{x}-1}=\frac{\sqrt{x}-1+2}{\sqrt{x}-1}=1+\frac{2}{\sqrt{x}-1}\)( ĐK : \(\hept{\begin{cases}x\ge0\\x\ne1\end{cases}}\))

Để A nguyên => \(\frac{2}{\sqrt{x}-1}\)nguyên

=> \(2⋮\sqrt{x}-1\)

=> \(\sqrt{x}-1\inƯ\left(2\right)=\left\{\pm1;\pm2\right\}\)

=> \(\sqrt{x}\in\left\{0;2;3\right\}\)< đã loại 1 trường hợp âm >

=> \(x\in\left\{0;4;9\right\}\)( tmđk )

Vậy với \(x\in\left\{0;4;9\right\}\)thì A có giá trị nguyên

\(pt\Leftrightarrow\sqrt{\left(x^4-9\right)+\left(x^3-3x\right)}+\sqrt{\left(x^4-9\right)+\left(2x^3-6x\right)}+\sqrt{x^2-3}=0\)

\(\Leftrightarrow\sqrt{\left(x^2-3\right)\left(x^2+x+3\right)}+\sqrt{\left(x^2-3\right)\left(x^2+2x+3\right)}+\sqrt{x^2-3}=0\)

\(\Leftrightarrow\sqrt{x^2-3}\left(\sqrt{x^2+x+3}+\sqrt{x^2+2x+3}+1\right)=0\)

\(\text{Nếu }x=\pm\sqrt{3}\Rightarrow\text{thỏa mãn còn lại thì thừa số số 2}>0\text{ nên không thỏa}\)