Editing Polynomial (section)

=== Multiplication ===
{{Further|Polynomial expansion}}
Polynomials can also be multiplied.  To expand the [[product (mathematics)|product]] of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other.<ref name="Edwards-1995-p47"/> For example, if
<math display="block">\begin{align}
 \color{Red} P &\color{Red}{= 2x + 3y + 5} \\
 \color{Blue} Q &\color{Blue}{= 2x + 5y + xy + 1}
\end{align}</math>
then
<math display="block">\begin{array}{rccrcrcrcr}
{\color{Red}{P}} {\color{Blue}{Q}} & {{=}}&&({\color{Red}{2x}}\cdot{\color{Blue}{2x}})
&+&({\color{Red}{2x}}\cdot{\color{Blue}{5y}})&+&({\color{Red}{2x}}\cdot {\color{Blue}{xy}})&+&({\color{Red}{2x}}\cdot{\color{Blue}{1}})
\\&&+&({\color{Red}{3y}}\cdot{\color{Blue}{2x}})&+&({\color{Red}{3y}}\cdot{\color{Blue}{5y}})&+&({\color{Red}{3y}}\cdot {\color{Blue}{xy}})&+&
({\color{Red}{3y}}\cdot{\color{Blue}{1}})
\\&&+&({\color{Red}{5}}\cdot{\color{Blue}{2x}})&+&({\color{Red}{5}}\cdot{\color{Blue}{5y}})&+&
({\color{Red}{5}}\cdot {\color{Blue}{xy}})&+&({\color{Red}{5}}\cdot{\color{Blue}{1}})
\end{array}</math>
Carrying out the multiplication in each term produces
<math display="block">\begin{array}{rccrcrcrcr}
PQ & = && 4x^2 &+& 10xy &+& 2x^2y &+& 2x \\
&&+& 6xy &+& 15y^2 &+& 3xy^2 &+& 3y \\
&&+& 10x &+& 25y &+& 5xy &+& 5.
\end{array}</math>
Combining similar terms yields
<math display="block">\begin{array}{rcccrcrcrcr}
PQ & = && 4x^2 &+&( 10xy + 6xy + 5xy ) &+& 2x^2y &+& ( 2x + 10x ) \\
&& + & 15y^2 &+& 3xy^2 &+&( 3y + 25y )&+&5
\end{array}</math>
which can be simplified to
<math display="block">PQ = 4x^2 + 21xy + 2x^2y + 12x + 15y^2 + 3xy^2 + 28y + 5.</math>
As in the example, the product of polynomials is always a polynomial.<ref name=":0" /><ref name=Barbeau-2003-pp1-2/>