Question

Subtract
${\mathrm{x}}^3+2{\mathrm{x}}^2\mathrm{y}+6{\mathrm{xy}}^2-{\mathrm{y}}^3\mathrm{from}{\mathrm{y}}^3-3{\mathrm{xy}}^2-4{\mathrm{x}}^2\mathrm{y}$

Answer 1

Subtraction of algebraic expressions

According to the question,

We have to subtract ${\mathrm{x}}^3+2{\mathrm{x}}^2\mathrm{y}+6{\mathrm{xy}}^2-{\mathrm{y}}^3\mathrm{from}{\mathrm{y}}^3-3{\mathrm{xy}}^2-4{\mathrm{x}}^2\mathrm{y}$

To subtract an algebraic expression from another, we should change the signs (from $\text{'}+\text{'}\mathrm{to}\text{'}-\text{'}$or from $\text{'}-\text{'}\mathrm{to}\text{'}+\text{'}$) of all the terms of the expression which is to be subtracted and then the two expressions are added.

$$\begin{gathered} {\mathrm{y}}^3-3{\mathrm{xy}}^2-4{\mathrm{x}}^2\mathrm{y}-({\mathrm{x}}^3+2{\mathrm{x}}^2\mathrm{y}+6{\mathrm{xy}}^2-{\mathrm{y}}^3) \\ ={\mathrm{y}}^3-3{\mathrm{xy}}^2-4{\mathrm{x}}^2\mathrm{y}-{\mathrm{x}}^3-2{\mathrm{x}}^2\mathrm{y}-6{\mathrm{xy}}^2+{\mathrm{y}}^3 \\ ={\mathrm{y}}^3+{\mathrm{y}}^3-3{\mathrm{xy}}^2-6{\mathrm{xy}}^2-4{\mathrm{x}}^2\mathrm{y}-2{\mathrm{x}}^2\mathrm{y}-{\mathrm{x}}^3 \\ =2{\mathrm{y}}^3-9{\mathrm{xy}}^2-6{\mathrm{x}}^2\mathrm{y}-{\mathrm{x}}^3 \end{gathered}$$

Hence, $2{\mathrm{y}}^3-9{\mathrm{xy}}^2-6{\mathrm{x}}^2\mathrm{y}-{\mathrm{x}}^3$ is the required answer.