Question

I hate polynomials so much thank you for your help!

Answer 1

Given the polynomials below;

For the first polynomial

`\left(5xy^2-3x^2y-2x+3xy\right)+\left(3xy^2+4x-5xy+2x^2y\right)`

Adding the polynomials gives

`\begin{gathered} \left(5xy^2-3x^2y-2x+3xy\right)+\left(3xy^2+4x-5xy+2x^2y\right)= \\ =5xy^2-3x^2y-2x+3xy+3xy^2+4x-5xy+2x^2y \\ \mathrm{Group\:like\:terms} \\ =5xy^2+3xy^2-3x^2y+2x^2y-2x+4x+3xy-5xy \\ =8xy^2-x^2y+2x-2xy \end{gathered}`

Hence, the matching polynomial is

`8xy^2-x^2y+2x-2xy`

For the second polynomial

`\left(4x^2y-3xy^2+4x-3xy\right)-\left(-4x^2y+2xy+3xy^2+x\right)`

Subtracting the polynomials gives

`\begin{gathered} \left(4x^2y-3xy^2+4x-3xy\right)-\left(-4x^2y+2xy+3xy^2+x\right) \\ =4x^2y-3xy^2+4x-3xy-\left(-4x^2y+2xy+3xy^2+x\right) \\ =4x^2y-3xy^2+4x-3xy+4x^2y-2xy-3xy^2-x \\ \mathrm{Group\:like\:terms} \\ =8x^2y-6xy^2+3x-5xy \end{gathered}`

Hence, the matching polynomial is

`8x^2y-6xy^2+3x-5xy`

For the third polynomial

`\left(2x-1\right)\left(4xy+3y^2-2y\right)`

Multiplying the polynomials

`\begin{gathered} =2x(\:4xy)+2x(3y^2)+2x\left(-2y\right)-1(\:4xy)-1(\:3y^2)-1\left(-2y\right) \\ =8x^2y+6xy^2-8xy-3y^2+2y \end{gathered}`

Hence, the matching polynomial is

`\begin{equation*} 8x^2y+6xy^2-8xy-3y^2+2y \end{equation*}`

For the fourth polynomial

`(16x^2y^3-2x^3y^2+4x^2y^2+4xy)/(2xy)`

Dividing the polynomials

`\begin{gathered} (16x^2y^3-2x^3y^2+4x^2y^2+4xy)/(2xy) \\ =(2xy\left(8xy^2-x^2y+2xy+2\right))/(2xy) \\ =8xy^2-x^2y+2xy+2 \end{gathered}`

Hence, the matching polynomial is

`\begin{equation*} 8xy^2-x^2y+2xy+2 \end{equation*}`