Multiplication of Polynomials
To multiply two or more monomials, we use the commutative and associative properties of multiplication along with the following properties of exponents.
Properties of Exponents
Let $$a$$ and $$b$$ denote real numbers. Then, if $$m$$ and $$n$$ are positive integers,
(1) $${a^m}{a^n} = {a^{m + n}}$$
(2) $${\left( {{a^m}} \right)^n} = {a^{mn}}$$
(3) $${\left( {ab} \right)^n} = {a^n}{b^n}$$
We verify (1) as follows:
\[{a^m}{a^n} = \overbrace {\left( {a \cdot a \cdots a} \right)}^{m{\text{ }}factors}\overbrace {\left( {a \cdot a \cdots a} \right)}^{n{\text{ }}factors} = \overbrace {a \cdot a \cdots a \cdot a \cdot a \cdots a}^{m + n{\text{ }}factors} = {a^{m + n}}\]
Properties (1), (2) and (3) are useful for simplifying algebraic expressions containing exponents. In general, when the properties of real numbers are used to rewrite an algebraic expression as compactly as possible or in a form so that further calculations are made easier, we say that the expression has been simplified. Its meaning is usually clear from the context in which it is used.
Example:
Use the properties of exponents to simplify each expression.
(a) $${x^5}{x^4}$$
(b) $$2{y^4}{y^6}{y^2}{z^2}$$
(c) $$\left( {5{x^3}y} \right)\left( { – 3{x^2}{y^4}} \right)$$
(d) $$\left( { – 2{x^4}{y^2}} \right)\left( {3{x^2}{y^3}} \right)\left( {5x{y^4}} \right)$$
(e) $${\left( {3{x^2}{y^4}} \right)^2}$$
Solution:
(a) $${x^5}{x^4} = {x^{5 + 4}} = {x^9}$$
(b) $$2{y^4}{y^6}{y^2}{z^2} = 2{y^{4 + 6 + 2}}{z^2} = 2{y^{12}}{z^2}$$
(c) $$\left( {5{x^3}y} \right)\left( { – 3{x^2}{y^4}} \right) = \left( 5 \right)\left( { – 3} \right)\left( {{x^3}{x^2}} \right)\left( {{y^1}{y^4}} \right)$$
$$ = – 15{x^5}{y^5}$$
(d) $$\left( { – 2{x^4}{y^2}} \right)\left( {3{x^2}{y^3}} \right)\left( {5x{y^4}} \right) = \left( { – 2} \right)\left( 3 \right)\left( 5 \right)\left( {{x^4}{x^2}x} \right)\left( {{y^2}{y^3}{y^4}} \right)$$
$$ = – 30{x^7}{y^9}$$
(e) $${\left( {3{x^2}{y^4}} \right)^2} = {3^2}{\left( {{x^2}} \right)^2}{\left( {{y^4}} \right)^2} = 9{x^{\left( 2 \right)\left( 2 \right)}}{y^{\left( 4 \right)\left( 2 \right)}}$$
$$ = 9{x^4}{y^8}$$
To multiply a polynomial by a monomial, we use the distributive property. Thus, we multiply each term of the polynomial by the monomial and then simplify the resulting products by using the properties of exponents.
Example:
Find the product of $$\left( { – 2{x^2}{y^3} + 5xy + 7} \right)\left( {4{x^3}y} \right)$$
Solution:
$$\left( { – 2{x^2}{y^3} + 5xy + 7} \right)\left( {4{x^3}y} \right) = \left( { – 3{x^2}{y^3}} \right)\left( {4{x^3}y} \right) + \left( {5xy} \right)\left( {4{x^3}y} \right) + 7\left( {4{x^3}y} \right)$$
$$ = – 12{x^5}{y^4} + 20{x^4}{y^2} + 28{x^3}y$$
To multiply two polynomials, we again employ the distribution property. Thus, we multiply each term of the first polynomial by each term of the second and combine like terms. This procedure is sometimes called expanding the products.
Example:
Expand the product $$\left( {{x^4} – 5{x^2} + 7} \right)\left( {3{x^2} + 2} \right)$$
Solution:
$$\left( {{x^4} – 5{x^2} + 7} \right)\left( {3{x^2} + 2} \right) = \left( {{x^4} – 5{x^2} + 7} \right)\left( {3{x^2}} \right) + \left( {{x^4} – 5{x^2} + 7} \right)\left( 2 \right)$$
$$ = \left( {3{x^6} – 15{x^4} + 21{x^2}} \right) + \left( {2{x^4} – 10{x^2} + 14} \right)$$
$$ = 3{x^6} – 13{x^4} + 11{x^2} + 14$$
Certain products of polynomials occur so often that it is useful to know the expanded forms by heart. The following list contains some of these special products.
Special Products
If $$a$$, $$b$$, $$c$$, $$d$$, $$x$$ and $$y$$ are real numbers, then:
- $$\left( {a – b} \right)\left( {a + b} \right) = {a^2} – {b^2}$$
- $${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$$
- $${\left( {a – b} \right)^2} = {a^2} – 2ab + {b^2}$$
- $$\left( {a + b} \right)\left( {{a^2} – ab + {b^2}} \right) = {a^3} + {b^3}$$
- $$\left( {a – b} \right)\left( {{a^2} + ab + {b^2}} \right) = {a^3} – {b^3}$$
- $${\left( {a + b} \right)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3}$$
- $${\left( {a – b} \right)^3} = {a^3} – 3{a^2}b + 3a{b^2} – {b^3}$$
- $$\left( {ax + by} \right)\left( {cx + dy} \right) = ac{x^2} + \left( {ad + bc} \right)xy + bd{y^2}$$
You should verify the expansions in the list above by actually doing the multiplication. They should become so familiar that you use them automatically in your calculations.