Bài 10:

a: \(x^2-36=0\)

=>\(x^2=36\)

=>\(\left[\begin{array}{l}x=6\\ x=-6\end{array}\right.\)

b: \(x^3-0,25x=0\)

=>\(x\left(x^2-0,25\right)=0\)

=>x(x-0,5)(x+0,5)=0

=>\(\left[\begin{array}{l}x=0\\ x-0,5=0\\ x+0,5=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=0\\ x=0,5\\ x=-0,5\end{array}\right.\)

c: \(\left(3x+2\right)^2-\left(x+1\right)^2=0\)

=>(3x+2-x-1)(3x+2+x+1)=0

=>(2x+1)(4x+3)=0

=>\(\left[\begin{array}{l}2x+1=0\\ 4x+3=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=-\frac12\\ x=-\frac34\end{array}\right.\)

d: \(49x^2-\left(5x-3\right)^2=0\)

=>\(\left(7x\right)^2-\left(5x-3\right)^2=0\)

=>(7x-5x+3)(7x+5x-3)=0

=>(2x+3)(12x-3)=0

=>\(\left[\begin{array}{l}2x+3=0\\ 12x-3=0\end{array}\right.\Rightarrow\left[\begin{array}{l}2x=-3\\ 12x=3\end{array}\right.\Rightarrow\left[\begin{array}{l}x=-\frac32\\ x=\frac{3}{12}=\frac14\end{array}\right.\)

Bài 9:

a: \(x^4+4\)

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

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

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

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

\(=\left(x+y-x+y\right)\left\lbrack\left(x+y\right)_{}^2+\left(x+y\right)\left(x-y\right)+\left(x-y\right)^2\right\rbrack\)

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

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

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

\(=\left(x^2+1\right)^2-\left(x\sqrt3\right)^2\)

\(=\left(x^2+1-x\sqrt3\right)\left(x^2+1+x\sqrt3\right)\)

Bài 8:

a: \(x^4-1\)

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

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

b: \(1-y^3+6xy^2-12x^2y+8x^3\)

\(=8x^3-12x^2y+6xy^2-y^3+1\)

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

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

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

c: \(x^4-y^2\left(2x-y\right)^2\)

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

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

d: \(\left(x+a\right)^4-\left(x-a\right)^4\)

\(=\left\lbrack\left(x+a\right)^2-\left(x-a\right)^2\right\rbrack\cdot\left\lbrack\left(x+a\right)^2+\left(x-a\right)^2\right\rbrack\)

\(=\left(x+a-x+a\right)\left(x+a+x-a\right)\left(x^2+2xa+a^2+x^2-2xa+a^2\right)\)

\(=2a\cdot2x\cdot\left(2x^2+2a^2\right)=8ax\left(x^2+a^2\right)\)

Bài 7:

a: \(x^4+2x^2y+y^2\)

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

b: \(\left(2x+y\right)^2-\left(x+2y\right)^2\)

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

=(3x+3y)(x-y)

=3(x+y)(x-y)

d: \(\left(8x^3-27y^3\right)-2x\left(4x^2-9y^2\right)\)

\(=8x^3-27y^3-8x^3+18xy^2=-27y^3+18xy^2\)

\(=-9y^2\left(3y-2x\right)\)

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

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

\(=\left(2x-1\right)\left(x^2+2x+1-x^2+x-2+x^2-4x+4\right)=\left(2x-1\right)\left(x^2-x+3\right)\)

f: \(64x^3+125y^3+5y\cdot\left(16x^2-25y^2\right)\)

\(=64x^3+125y^3+80x^2y-125y^3=64x^3+80x^2y\)

\(=16x^2\left(4x+5y\right)\)