# Multiplying Polynomials by Monomial

To multiply polynomial by monomial, one should use distributive property of multiplication.

Then, just multiply monomials and you're done.

Example 1. Multiply ${\left({2}{x}\right)}{\left({3}{{x}}^{{2}}+{5}{x}+{4}\right)}$.

${\color{red}{{{\left({2}{x}\right)}}}}{\color{green}{{{\left({3}{{x}}^{{2}}+{5}{x}+{4}\right)}}}}=$

$={\color{red}{{{\left({2}{x}\right)}}}}\cdot{\color{green}{{{\left({3}{{x}}^{{2}}\right)}}}}+{\color{red}{{{\left({2}{x}\right)}}}}\cdot{\color{green}{{{\left({5}{x}\right)}}}}+{\color{red}{{{\left({2}{x}\right)}}}}\cdot{\color{green}{{{\left({4}\right)}}}}=$ (distributive property of multiplication)

$={6}{{x}}^{{3}}+{10}{{x}}^{{2}}+{8}{x}$ (multiply monomials)

Answer: ${\left({2}{x}\right)}{\left({3}{{x}}^{{2}}+{5}{x}+{4}\right)}={6}{{x}}^{{3}}+{10}{{x}}^{{2}}+{8}{x}$.

Negative terms are handled in the same way.

Example 2. Multiply the following: ${\left({{x}}^{{3}}-{5}{{x}}^{{2}}-{x}+{7}\right)}\frac{{1}}{{3}}{{x}}^{{2}}$.

${\color{green}{{{\left({{x}}^{{3}}-{5}{{x}}^{{2}}-{x}+{7}\right)}}}}{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}=$

$={\color{green}{{{{x}}^{{3}}}}}\cdot{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}+{\color{green}{{{\left(-{5}{{x}}^{{2}}\right)}}}}\cdot{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}+{\color{green}{{{\left(-{x}\right)}}}}\cdot{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}+{\color{green}{{{7}}}}\cdot{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}=$ (distributive property of multiplication)

$=\frac{{1}}{{3}}{{x}}^{{5}}-\frac{{5}}{{3}}{{x}}^{{4}}-\frac{{1}}{{3}}{{x}}^{{3}}+\frac{{7}}{{3}}{{x}}^{{2}}$ (multiply monomials)

Answer: ${\left({{x}}^{{3}}-{5}{{x}}^{{2}}-{x}\right)}\frac{{1}}{{3}}{{x}}^{{2}}=\frac{{1}}{{3}}{{x}}^{{5}}-\frac{{5}}{{3}}{{x}}^{{4}}-\frac{{1}}{{3}}{{x}}^{{3}}+\frac{{7}}{{3}}{{x}}^{{2}}$.

Of course, polynomials with many variables can also be handled in a similar way.

Example 3. Multiply $-{3}{x}{{y}}^{{2}}$ by ${\left({3}{{x}}^{{2}}{y}+{2}{x}{z}-{5}{x}{{y}}^{{2}}-{z}\right)}$.

$-{3}{x}{{y}}^{{2}}{\left({3}{{x}}^{{2}}{y}+{2}{x}{z}-{5}{x}{{y}}^{{2}}-{z}\right)}=$

$={\left(-{3}{x}{{y}}^{{2}}\right)}{\left({3}{{x}}^{{2}}{y}\right)}+{\left(-{3}{x}{{y}}^{{2}}\right)}{\left({2}{x}{z}\right)}+{\left(-{3}{x}{{y}}^{{2}}\right)}{\left(-{5}{x}{{y}}^{{2}}\right)}+{\left(-{3}{x}{{y}}^{{2}}\right)}{\left(-{z}\right)}=$

$=-{9}{{x}}^{{3}}{{y}}^{{3}}-{6}{{x}}^{{2}}{{y}}^{{2}}{z}+{15}{{x}}^{{2}}{{y}}^{{4}}+{3}{x}{{y}}^{{2}}{z}$.

Answer: $-{3}{x}{{y}}^{{2}}{\left({3}{{x}}^{{2}}{y}+{2}{x}{z}-{5}{x}{{y}}^{{2}}-{z}\right)}=-{9}{{x}}^{{3}}{{y}}^{{3}}-{6}{{x}}^{{2}}{{y}}^{{2}}{z}+{15}{{x}}^{{2}}{{y}}^{{4}}+{3}{x}{{y}}^{{2}}{z}$.

Now, it is time to exercise.

Exercise 1. Multiply ${\left({{x}}^{{3}}+{2}{x}+{4}\right)}\cdot{\left({5}{{x}}^{{2}}\right)}$.

Answer: ${5}{{x}}^{{5}}+{10}{{x}}^{{3}}+{20}{{x}}^{{2}}$.

Exercise 2. Multiply $-\frac{{2}}{{7}}{{a}}^{{3}}{\left({{a}}^{{3}}-{2}{{a}}^{{2}}+{7}{b}\right)}$.

Answer: $-\frac{{2}}{{7}}{{a}}^{{6}}+\frac{{4}}{{7}}{{a}}^{{5}}-{2}{{a}}^{{3}}{b}$.

Exercise 3. Multiply ${\left(-{3}{a}{b}\right)}{\left({5}{{a}}^{{2}}{b}-{3}{{a}}^{{3}}{b}{c}+\frac{{3}}{{5}}{{a}}^{{2}}{{b}}^{{2}}-\frac{{1}}{{10}}{a}{b}\right)}$.

Answer: $-{15}{{a}}^{{3}}{{b}}^{{2}}+{9}{{a}}^{{4}}{{b}}^{{2}}{c}-\frac{{9}}{{5}}{{a}}^{{3}}{{b}}^{{3}}+\frac{{3}}{{10}}{{a}}^{{2}}{{b}}^{{2}}$.