Dividing Polynomials by Monomial

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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: $\frac{{5}}{{3}}{a}{b}-{{a}}^{{2}}{c}+\frac{{1}}{{5}}{a}-\frac{{1}}{{30}}$.