PDF questions

i) Division of a monomial by another monomial:

A monomial $40x{y}^{2}$ is divided by another monomial $10y$ will result in $\frac{40x{y}^{2}}{10y}=\frac{\overline{)10}×4×x×y×\overline{)y}}{\overline{)10}×\overline{)y}}=4\mathit{xy}$.

The result of dividing a monomial by another monomial will be a monomial.

ii) Division of a polynomial by a monomial:

Divide each term of the polynomial by the monomial to get the result of the division.

A polynomial $-12\mathit{xy}{z}^{3}+60$ is divided by a monomial $4z$ will result in:

$\frac{-12\mathit{xy}{z}^{3}+60}{4z}=\frac{-12\mathit{xy}{z}^{3}}{4z}+\frac{60}{4z}$.

$\begin{array}{l}=\frac{{\overline{)-12}}^{-3}×x×y×z×z×\overline{)z}}{\overline{)4}×\overline{)z}}+{\frac{\overline{)60}}{\overline{)4}z}}^{15}\\ \\ =-3\mathit{xy}{z}^{2}+\frac{15}{z}\end{array}$

Dividing any polynomial by a monomial will result in a polynomial.

The relation between the power of exponents and division of an algebraic expression by another algebraic expression:

$\frac{{a}^{n}}{{a}^{m}}={a}^{n}-{a}^{m},n>m,a\ne 0.$
$\frac{4x{y}^{2}}{2y}=2x{y}^{2-1}$
$=2\mathit{xy}$
When a monomial is divided by itself, we will get $$1$$.