# Multiplying Polynomials

To multiply polynomials, you need to multiply each term of first polynomial by each term of another polynomial, then add resulting products, simplify and combine like terms (if possible).

As was stated in FOIL note, FOIL is not applicable for multiplying general polynomials (only binomials, which are polynomials with two terms).

But technique is still the same.

Above rule follows from multiple applications of distributive property of multiplication.

It appears, that FOIL is the particular case of above rule.

Let's see on example, how this works.

Example 1. Multiply ${\left({2}{{x}}^{{2}}+{7}\right)}{\left({{x}}^{{2}}+{3}{x}+{5}\right)}$.

If we treat ${\left({{x}}^{{2}}+{3}{x}+{5}\right)}$ as single symbol, then, according to distributive property of multiplication: ${\color{red}{{{\left({2}{{x}}^{{2}}+{7}\right)}}}}{\color{green}{{{\left({{x}}^{{2}}+{3}{x}+{5}\right)}}}}={\color{red}{{{2}{{x}}^{{2}}}}}{\color{green}{{{\left({{x}}^{{2}}+{3}{x}+{5}\right)}}}}+{\color{red}{{{7}}}}{\color{green}{{{\left({{x}}^{{2}}+{3}{x}+{5}\right)}}}}=$

$={2}{{x}}^{{2}}\cdot{{x}}^{{2}}+{2}{{x}}^{{2}}\cdot{3}{x}+{2}{{x}}^{{2}}\cdot{5}+{7}\cdot{{x}}^{{2}}+{7}\cdot{3}{x}+{7}\cdot{5}$ (multiply monomial by polynomial)

$={2}{{x}}^{{4}}+{6}{{x}}^{{3}}+{10}{{x}}^{{2}}+{7}{{x}}^{{2}}+{21}{x}+{35}$ (multiply monomials)

$={2}{{x}}^{{4}}+{6}{{x}}^{{3}}+{17}{{x}}^{{2}}+{21}{x}+{35}$ (combine like terms)

Answer: ${\left({2}{{x}}^{{2}}+{7}\right)}{\left({{x}}^{{2}}+{3}{x}+{5}\right)}={2}{{x}}^{{4}}+{6}{{x}}^{{3}}+{17}{{x}}^{{2}}+{21}{x}+{35}$.

As can be seen from Example 1, we multiplied each term of the first polynomial by each term of the second polynomial.

Now, we can use this rule.

As always, you need to be careful with minus signs.

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

Multiply each term of the first polynomial by each term of second polynomial:

${\left({\color{ma\genta}{{{\left(-{2}{a}{b}\right)}}}}+{\color{red}{{{3}{b}{c}}}}+{\color{green}{{{2}}}}\right)}{\left({\color{blue}{{{{a}}^{{2}}{{b}}^{{2}}}}}+{\color{brown}{{{a}{b}}}}+{\color{purple}{{{\left(-{7}{c}\right)}}}}\right)}=$

$={\color{ma\genta}{{{\left(-{2}{a}{b}\right)}}}}\cdot{\color{blue}{{{{a}}^{{2}}{{b}}^{{2}}}}}+{\color{ma\genta}{{{\left(-{2}{a}{b}\right)}}}}\cdot{\color{brown}{{{a}{b}}}}+{\color{ma\genta}{{{\left(-{2}{a}{b}\right)}}}}\cdot{\color{purple}{{{\left(-{7}{c}\right)}}}}+$

$+\ \ \ {\color{red}{{{3}{b}{c}}}}\cdot{\color{blue}{{{{a}}^{{2}}{{b}}^{{2}}}}}+{\color{red}{{{3}{b}{c}}}}\cdot{\color{brown}{{{a}{b}}}}+{\color{red}{{{3}{b}{c}}}}\cdot{\color{purple}{{{\left(-{7}{c}\right)}}}}+$

$+\ \ \ {\color{green}{{{2}}}}\cdot{\color{blue}{{{{a}}^{{2}}{{b}}^{{2}}}}}+{\color{green}{{{2}}}}\cdot{\color{brown}{{{a}{b}}}}+{\color{green}{{{2}}}}\cdot{\color{purple}{{{\left(-{7}{c}\right)}}}}$

$=-{2}{{a}}^{{3}}{{b}}^{{3}}-{2}{{a}}^{{2}}{{b}}^{{2}}+{14}{a}{b}{c}+{3}{{a}}^{{2}}{{b}}^{{3}}{c}+{3}{a}{{b}}^{{2}}{c}-{21}{b}{{c}}^{{2}}+{2}{{a}}^{{2}}{{b}}^{{2}}+{2}{a}{b}-{14}{c}$ (multiply monomials)

$=-{2}{{a}}^{{3}}{{b}}^{{3}}+{14}{a}{b}{c}+{3}{{a}}^{{2}}{{b}}^{{3}}{c}+{3}{a}{{b}}^{{2}}{c}-{21}{b}{{c}}^{{2}}+{2}{a}{b}-{14}{c}$ (combine like terms)

Answer: ${\left(-{2}{a}{b}+{3}{b}{c}+{2}\right)}{\left({{a}}^{{2}}{{b}}^{{2}}+{a}{b}-{7}{c}\right)}=-{2}{{a}}^{{3}}{{b}}^{{3}}+{14}{a}{b}{c}+{3}{{a}}^{{2}}{{b}}^{{3}}{c}+{3}{a}{{b}}^{{2}}{c}-{21}{b}{{c}}^{{2}}+{2}{a}{b}-{14}{c}$.

Of course, you can multiply more than two polynomials. For this, you, gradually, need to apply above rule more than 1 time.

Multiply first two polynomials. Product of polynomials is polynomial itself, so you will get polynomial.

Next, multiply resulting polynomial by the next polynomial etc.

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

Multiply first two polynomials: ${\left({4}{{x}}^{{2}}-{4}{x}+{1}\right)}{\left({2}{x}-{3}\right)}={8}{{x}}^{{3}}-{20}{{x}}^{{2}}+{14}{x}-{3}$.

Next, multiply resulting polynomial by the third polynomial:

${\left({8}{{x}}^{{3}}-{20}{{x}}^{{2}}+{14}{x}-{3}\right)}{\left(-{2}+{5}{x}-{{x}}^{{2}}\right)}=-{8}{{x}}^{{5}}+{60}{{x}}^{{4}}-{130}{{x}}^{{3}}+{113}{{x}}^{{2}}-{43}{x}+{6}$.

Answer: ${\left({4}{{x}}^{{2}}-{4}{x}+{1}\right)}{\left({2}{x}-{3}\right)}{\left(-{2}+{5}{x}-{{x}}^{{2}}\right)}=-{8}{{x}}^{{5}}+{60}{{x}}^{{4}}-{130}{{x}}^{{3}}+{113}{{x}}^{{2}}-{43}{x}+{6}$.

Now, it is time to exercise.

Exercise 1. Multiply ${\left({2}{x}+{3}{y}+{7}{z}\right)}{\left({3}{x}-{5}{z}-{2}{y}\right)}$.

Answer: ${6}{{x}}^{{2}}+{5}{x}{y}+{11}{x}{z}-{6}{{y}}^{{2}}-{29}{y}{z}-{35}{{z}}^{{2}}$.

Exercise 2. Multiply ${\left(-\frac{{1}}{{3}}{a}{b}+{2}{b}{c}\right)}{\left({{a}}^{{2}}{{b}}^{{2}}-{9}{b}{c}-{3}{a}{b}+\frac{{2}}{{7}}{{c}}^{{2}}\right)}$.

Answer: $-\frac{{1}}{{3}}{{a}}^{{3}}{{b}}^{{3}}+{2}{{a}}^{{2}}{{b}}^{{3}}{c}+{{a}}^{{2}}{{b}}^{{2}}-{3}{a}{{b}}^{{2}}{c}-\frac{{2}}{{21}}{a}{b}{{c}}^{{2}}-{18}{{b}}^{{2}}{{c}}^{{2}}+\frac{{4}}{{7}}{b}{{c}}^{{3}}$.

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

Answer: ${{x}}^{{6}}-{2}{{x}}^{{5}}+{2}{{x}}^{{3}}-{2}{{x}}^{{2}}+{1}$.