Dividing Polynomials by Monomial
To multiply polynomial by monomial, one should use sum rule for fractions in reverse direction: $$$\frac{{{a}+{b}}}{{c}}=\frac{{a}}{{c}}+\frac{{b}}{{c}}$$$ (in fact, there can be more than two terms in the numerator).
In other words, we just break down polynomial, then, just divide monomials and you're done.
Example 1. Divide $$$\frac{{{4}{{x}}^{{3}}+{6}{{x}}^{{2}}+{10}{x}}}{{{2}{x}}}$$$.
$$$\frac{{\color{green}{{{4}{{x}}^{{3}}+{6}{{x}}^{{2}}+{10}{x}}}}}{{{\color{red}{{{2}{x}}}}}}=$$$
$$$=\frac{{{\color{green}{{{4}{{x}}^{{3}}}}}}}{{{\color{red}{{{2}{x}}}}}}+\frac{{{\color{green}{{{6}{{x}}^{{2}}}}}}}{{{\color{red}{{{2}{x}}}}}}+\frac{{{\color{green}{{{10}{x}}}}}}{{{\color{red}{{{2}{x}}}}}}=$$$ (break down polynomial)
$$$={2}{{x}}^{{2}}+{3}{x}+{5}$$$ (divide monomials)
Answer: $$$\frac{{{4}{{x}}^{{3}}+{6}{{x}}^{{2}}+{10}{x}}}{{{2}{x}}}={2}{{x}}^{{2}}+{3}{x}+{5}$$$.
Negative terms are handled in the same way.
Example 2. Multiply the following: $$${\left({3}{{x}}^{{3}}-{{x}}^{{2}}-{5}{x}+\frac{{3}}{{7}}\right)}\div{\left(\frac{{1}}{{3}}{{x}}^{{2}}\right)}$$$.
$$$\frac{{{\color{green}{{{2}{{x}}^{{3}}-{{x}}^{{2}}-{5}{x}+\frac{{3}}{{7}}}}}}}{{{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}}}=$$$
$$$=\frac{{{\color{green}{{{2}{{x}}^{{3}}}}}}}{{{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}}}+\frac{{{\color{green}{{-{{x}}^{{2}}}}}}}{{{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}}}+\frac{{{\color{green}{{-{5}{x}}}}}}{{{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}}}+\frac{{{\color{green}{{\frac{{3}}{{7}}}}}}}{{{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}}}=$$$ (split polynomial)
$$$={6}{x}+{3}-\frac{{15}}{{x}}+\frac{{9}}{{{7}{{x}}^{{2}}}}$$$ (divide monomials)
Answer: $$${\left({3}{{x}}^{{3}}-{{x}}^{{2}}-{5}{x}+\frac{{3}}{{7}}\right)}\div{\left(\frac{{1}}{{3}}{{x}}^{{2}}\right)}={6}{x}+{3}-\frac{{15}}{{x}}+\frac{{9}}{{{7}{{x}}^{{2}}}}$$$.
Note: above example shows, that result of division polynomial by monomial is not always polynomial.
Of course, polynomials with many variables can also be handled in a similar way.
Example 3. Divide $$${\left({3}{{x}}^{{5}}{y}+{2}{x}{z}-{7}{x}{{y}}^{{2}}-{z}\right)}$$$ by $$$-{4}{{x}}^{{2}}{{y}}^{{2}}$$$.
$$$\frac{{{3}{{x}}^{{5}}{y}+{2}{x}{z}-{7}{x}{{y}}^{{2}}-{z}}}{{-{4}{{x}}^{{2}}{{y}}^{{2}}}}=$$$
$$$=\frac{{{3}{{x}}^{{5}}{y}}}{{-{4}{{x}}^{{2}}{{y}}^{{2}}}}+\frac{{{2}{x}{z}}}{{-{4}{{x}}^{{2}}{{y}}^{{2}}}}+\frac{{-{7}{x}{{y}}^{{2}}}}{{-{4}{{x}}^{{2}}{{y}}^{{2}}}}+\frac{{-{z}}}{{-{4}{{x}}^{{2}}{{y}}^{{2}}}}=$$$
$$$=-\frac{{{3}{{x}}^{{3}}}}{{{4}{y}}}-\frac{{z}}{{{2}{x}{{y}}^{{2}}}}+\frac{{7}}{{{4}{x}}}+\frac{{z}}{{{4}{{x}}^{{2}}{{y}}^{{2}}}}$$$.
Answer: $$$\frac{{{3}{{x}}^{{5}}{y}+{2}{x}{z}-{7}{x}{{y}}^{{2}}-{z}}}{{-{4}{{x}}^{{2}}{{y}}^{{2}}}}=-\frac{{{3}{{x}}^{{3}}}}{{{4}{y}}}-\frac{{z}}{{{2}{x}{{y}}^{{2}}}}+\frac{{7}}{{{4}{x}}}+\frac{{z}}{{{4}{{x}}^{{2}}{{y}}^{{2}}}}$$$.
Now, it is time to exercise.
Exercise 1. Divide $$$\frac{{{{x}}^{{5}}+{2}{{x}}^{{4}}+{5}{{x}}^{{2}}}}{{{5}{{x}}^{{2}}}}$$$.
Answer: $$$\frac{{1}}{{5}}{{x}}^{{3}}+\frac{{2}}{{5}}{{x}}^{{2}}+{1}$$$.
Exercise 2. Divide $$${\left({3}{{a}}^{{3}}-{7}{{a}}^{{2}}+{2}{b}\right)}$$$ by $$$-\frac{{2}}{{7}}{{a}}^{{2}}$$$.
Answer: $$$-\frac{{21}}{{2}}{a}+\frac{{49}}{{2}}-\frac{{{7}{b}}}{{{{a}}^{{2}}}}$$$.
Exercise 3. Divide the following: $$${\left({5}{{a}}^{{2}}{{b}}^{{2}}-{3}{{a}}^{{3}}{b}{c}+\frac{{3}}{{5}}{{a}}^{{2}}{b}-\frac{{1}}{{10}}{a}{b}\right)}\div{\left({3}{a}{b}\right)}$$$.
Answer: $$$\frac{{5}}{{3}}{a}{b}-{{a}}^{{2}}{c}+\frac{{1}}{{5}}{a}-\frac{{1}}{{30}}$$$.