b)\(\frac{9x^4-6x^3+15x^2+2x+1}{3x^2-2x+5}=\frac{3x^2.\left(3x^2-2x+5\right)+2x+1}{3x^2-2x+5}=3x^2+\frac{2x+1}{3x^2-2x+5}\)

=> đa thức dư trong phép chia là 2x+1

\(\frac{x^3+2x^2-3x+9}{x+3}=\frac{x^3+9x^2+27x+27-7x^2-30x-18}{x+3}=\frac{\left(x+3\right)^3-7x^2-30x-18}{x+3}\)

\(\left(x+3\right)^2-\frac{7x^2+21x+9x+18}{x+3}=\left(x+3\right)^2-\frac{7x.\left(x+3\right)+9.\left(x+3\right)-9}{x+3}\)

\(=\left(x+3\right)^2-\frac{\left(7x+9\right).\left(x+3\right)-9}{x+3}=\left(x+3\right)^2-\left(7x+9\right)-\frac{9}{x+3}\)

=> đa thức dư trong phép chia là 9

p/s: t mới lớp 7_sai sót mong bỏ qua :>

a) Ta có: \(\left(\dfrac{1}{x^2+x}-\dfrac{2-x}{x+1}\right):\left(\dfrac{1}{x}+x-2\right)\)

\(=\left(\dfrac{1}{x\left(x+1\right)}+\dfrac{x+2}{x+1}\right):\left(\dfrac{1}{x}+x-2\right)\)

\(=\dfrac{x^2+2x+1}{x\left(x+1\right)}:\dfrac{x^2-2x+1}{x}\)

\(=\dfrac{\left(x+1\right)^2}{x\left(x+1\right)}\cdot\dfrac{x}{\left(x-1\right)^2}\)

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

b) Ta có: \(\left(\dfrac{3x}{1-3x}+\dfrac{2x}{3x+1}\right):\dfrac{6x^2+10x}{1-6x+9x^2}\)

\(=\dfrac{3x\left(3x+1\right)+2x\left(1-3x\right)}{\left(1-3x\right)\left(1+3x\right)}:\dfrac{2x\left(3x+5\right)}{\left(1-3x\right)^2}\)

\(=\dfrac{9x^2+3x+2x-6x^2}{\left(1-3x\right)\left(1+3x\right)}:\dfrac{2x\left(3x+5\right)}{\left(1-3x\right)^2}\)

\(=\dfrac{3x^2+5x}{\left(1-3x\right)\left(1+3x\right)}\cdot\dfrac{\left(1-3x\right)^2}{2x\left(3x+5\right)}\)

\(=\dfrac{x\left(3x+5\right)}{1+3x}\cdot\dfrac{1-3x}{2x\left(3x+5\right)}\)

\(=\dfrac{2\left(1-3x\right)}{3x+1}\)

c) Ta có: \(\left(\dfrac{9}{x^3-9x}+\dfrac{1}{x+3}\right):\left(\dfrac{x-3}{x^2+3x}-\dfrac{x}{3x+9}\right)\)

\(=\left(\dfrac{9}{x\left(x-3\right)\left(x+3\right)}+\dfrac{1}{x+3}\right):\left(\dfrac{x-3}{x\left(x+3\right)}-\dfrac{x}{3\left(x+3\right)}\right)\)

\(=\dfrac{9+x\left(x-3\right)}{x\left(x-3\right)\left(x+3\right)}:\dfrac{3\left(x-3\right)-x^2}{3x\left(x+3\right)}\)

\(=\dfrac{9+x^2-3x}{x\left(x-3\right)\left(x+3\right)}\cdot\dfrac{3x\left(x+3\right)}{3x-9-x^2}\)

\(=\dfrac{x^2-3x+9}{x-3}\cdot\dfrac{3}{-\left(x^2-3x+9\right)}\)

\(=\dfrac{-3}{x-3}\)

\(3x^n\left(6x^{n-3}+1\right)-2x^n\left(9x^{n-3}-1\right)\)

\(=18x^{2n-3}+3x^n-18x^{2n-3}+2x^n\)

\(=3x^n+2x^n\)

\(=5x^n\)

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

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

\(=2x^3y-6x^2y^2+6x^2y^2-18xy^3\)

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

b) \(\left(6x^5y^2-9x^4y^3+15x^3y^4\right):3x^3y^2\)

\(=6x^5y^2:3x^3y^2-9x^4y^3:3x^3y^2+15x^3y^4:3x^3y^2\)

\(=2x^2-3xy+5y^2\)

c) \(\left(2x+3\right)^2+\left(2x+5\right)^2-2\left(2x+3\right)\left(2x+5\right)\)

\(=\left(2x+3-2x-5\right)^2\)

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

d) \(\left(y+3\right)^3-\left(3-y\right)^2-54y\)

\(=y^3+9y^2+27y+27-\left(x^2-6x+9\right)-54y\)

\(=y^3+9y^2-27y+27-x^2+6y-9\)

\(=y^3+9y^2-x^2-21y+18\)

a: \(=3y^2-5x^2y^3-2y^2+3x^2y^3=y^2-2x^2y^3\)

b: \(=6x-y+2x^2+3y^2-2x^2+x=7x-y+3y^2\)

c: \(=x-y+4y^2-6xy+\dfrac{10x^2}{y}\)

\(a.\left(9x^2y^3-15x^4y^4\right):3x^2y-\left(2-3x^2y\right)y^2\)

\(=3y^2-5x^2y^3-2y^2+3x^2y^3\)

\(=y^2-2x^2y^3\)

\(b.\left(6x^2-xy\right):x+\left(2x^3y+3xy^2\right):xy-\left(2x-1\right)x\)

\(=6x-y+2x^2+3y-2+x\)

\(=2x^2+7x+2y-2\)

\(c.\left(x^2-xy\right):x+\left(6x^2y^5-9x^3y^4+15x^4y^3\right):\dfrac{3}{2}x^2y^3\)

\(=x-y+4y^2-6xy+10x^2\)

a: =2x^5-15x^3-x^2-2x^5-x^3=-16x^3-x^2

b: =x^3+3x^2-2x-3x^2-9x+6

=x^3-11x+6

c: \(=\dfrac{4x^3+2x^2-6x^2-3x-2x-1+5}{2x+1}\)

\(=2x^2-3x-1+\dfrac{5}{2x+1}\)

a) \(6x^3\left(\dfrac{1}{3}x^2-\dfrac{5}{2}-\dfrac{1}{6}\right)-2x^5-x^3\)

\(=6x^3\left(\dfrac{1}{3}x^2-\dfrac{16}{6}\right)-2x^5-x^3\)

\(=2x^5-16x^3-2x^5-x^3\)

\(=-17x^3\)

b) \(\left(x+3\right)\left(x^2+3x-2\right)\)

\(=x^3+3x^2-2x+3x^2+9x-6\)

\(=x^3+6x^2+7x-6\)

c) \(\left(4x^3-4x^2-5x+4\right):\left(2x+1\right)\)

\(=2x^2+4x^3-2x-4x^2-\dfrac{5}{2}-5x+\dfrac{2}{x}+4\)

\(=4x^3-2x^2-7x+\dfrac{2}{x}+\dfrac{3}{2}\)