Question

Complete the table of products.
$\begin{array}{*{20}{c}}
  {\dfrac{{{1^{st}}monomial \to }}{{{2^{nd}}monomial \downarrow }}}&{2x}&{ - 5y}&{3{x^2}}&{ - 4xy}&{7{x^2}y}&{ - 9{x^2}{y^2}} \\
  {2x}&{4{x^2}}& - & - & - & - & - \\
  { - 5y}& - & - &{ - 15{x^2}y}& - & - & - \\
  {3{x^2}}& - & - & - & - & - & - \\
  { - 4xy}& - & - & - & - & - & - \\
  {7{x^2}y}& - & - & - & - & - & - \\
  { - 9{x^2}{y^2}}& - & - & - & - & - & -
\end{array}$

Answer 1

Hint: When terms with similar variables (exponent>0) irrespective of their exponents are multiplied, then the exponent of the result must be greater than the multiplicands. Anything multiplied with zero is zero.

Complete step-by-step solution
Multiplying 2x with rest of the terms starting from -5y
$
  2x \times \left( { - 5y} \right) = - 10xy \\
  2x \times 3{x^2} = 6{x^3} \\
  2x \times \left( { - 4xy} \right) = - 8{x^2}y \\
  2x \times 7{x^2}y = 14{x^3}y \\
  2x \times \left( { - 9{x^2}{y^2}} \right) = - 18{x^3}{y^2} \\
 $
Fill the second row with the above values
$\begin{array}{*{20}{c}}
  {\dfrac{{{1^{st}}monomial \to }}{{{2^{nd}}monomial \downarrow }}}&{2x}&{ - 5y}&{3{x^2}}&{ - 4xy}&{7{x^2}y}&{ - 9{x^2}{y^2}} \\
  {2x}&{4{x^2}}&{ - 10xy}&{6{x^3}}&{ - 8{x^2}y}&{14{x^3}y}&{ - 18{x^3}{y^2}} \\
  { - 5y}& - & - &{ - 15{x^2}y}& - & - & - \\
  {3{x^2}}& - & - & - & - & - & - \\
  { - 4xy}& - & - & - & - & - & - \\
  {7{x^2}y}& - & - & - & - & - & - \\
  { - 9{x^2}{y^2}}& - & - & - & - & - & -
\end{array}$
Multiplying -5y with rest of the terms starting from 2x except $3{x^2}$
$
   - 5y \times 2x = - 10xy \\
   - 5y \times \left( { - 5y} \right) = 25{y^2} \\
   - 5y \times 3{x^2} = - 15{x^2}y \\
   - 5y \times \left( { - 4xy} \right) = 20x{y^2} \\
   - 5y \times 7{x^2}y = - 35{x^2}{y^2} \\
   - 5y \times \left( { - 9{x^2}{y^2}} \right) = 45{x^2}{y^3} \\
 $
Fill the third row with the above values
$\begin{array}{*{20}{c}}
  {\dfrac{{{1^{st}}monomial \to }}{{{2^{nd}}monomial \downarrow }}}&{2x}&{ - 5y}&{3{x^2}}&{ - 4xy}&{7{x^2}y}&{ - 9{x^2}{y^2}} \\
  {2x}&{4{x^2}}&{ - 10xy}&{6{x^3}}&{ - 8{x^2}y}&{14{x^3}y}&{ - 18{x^3}{y^2}} \\
  { - 5y}&{ - 10xy}&{25{y^2}}&{ - 15{x^2}y}&{20x{y^2}}&{ - 35{x^2}{y^2}}&{45{x^2}{y^3}} \\
  {3{x^2}}& - & - & - & - & - & - \\
  { - 4xy}& - & - & - & - & - & - \\
  {7{x^2}y}& - & - & - & - & - & - \\
  { - 9{x^2}{y^2}}& - & - & - & - & - & -
\end{array}$
Multiplying $3{x^2}$ with rest of the terms starting from 2x
$
  3{x^2} \times 2x = 6{x^3} \\
  3{x^2} \times \left( { - 5y} \right) = - 15{x^2}y \\
  3{x^2} \times 3{x^2} = 9{x^4} \\
  3{x^2} \times \left( { - 4xy} \right) = - 12{x^3}y \\
  3{x^2} \times 7{x^2}y = 21{x^4}y \\
  3{x^2} \times \left( { - 9{x^2}{y^2}} \right) = - 27{x^4}{y^2} \\
 $
Fill the fourth row with the above values
$\begin{array}{*{20}{c}}
  {\dfrac{{{1^{st}}monomial \to }}{{{2^{nd}}monomial \downarrow }}}&{2x}&{ - 5y}&{3{x^2}}&{ - 4xy}&{7{x^2}y}&{ - 9{x^2}{y^2}} \\
  {2x}&{4{x^2}}&{ - 10xy}&{6{x^3}}&{ - 8{x^2}y}&{14{x^3}y}&{ - 18{x^3}{y^2}} \\
  { - 5y}&{ - 10xy}&{25{y^2}}&{ - 15{x^2}y}&{20x{y^2}}&{ - 35{x^2}{y^2}}&{45{x^2}{y^3}} \\
  {3{x^2}}&{6{x^3}}&{ - 15{x^2}y}&{9{x^4}}&{ - 12{x^3}y}&{21{x^4}y}&{ - 27{x^4}{y^2}} \\
  { - 4xy}& - & - & - & - & - & - \\
  {7{x^2}y}& - & - & - & - & - & - \\
  { - 9{x^2}{y^2}}& - & - & - & - & - & -
\end{array}$
Multiplying $ - 4xy$ with rest of the terms starting from 2x
$
   - 4xy \times 2x = - 8{x^2}y \\
   - 4xy \times \left( { - 5y} \right) = 20x{y^2} \\
   - 4xy \times 3{x^2} = - 12{x^3}y \\
   - 4xy \times \left( { - 4xy} \right) = 16{x^2}{y^2} \\
   - 4xy \times 7{x^2}y = - 28{x^3}{y^2} \\
   - 4xy \times \left( { - 9{x^2}{y^2}} \right) = 36{x^3}{y^3} \\
 $
Fill the fifth row with the above values
$\begin{array}{*{20}{c}}
  {\dfrac{{{1^{st}}monomial \to }}{{{2^{nd}}monomial \downarrow }}}&{2x}&{ - 5y}&{3{x^2}}&{ - 4xy}&{7{x^2}y}&{ - 9{x^2}{y^2}} \\
  {2x}&{4{x^2}}&{ - 10xy}&{6{x^3}}&{ - 8{x^2}y}&{14{x^3}y}&{ - 18{x^3}{y^2}} \\
  { - 5y}&{ - 10xy}&{25{y^2}}&{ - 15{x^2}y}&{20x{y^2}}&{ - 35{x^2}{y^2}}&{45{x^2}{y^3}} \\
  {3{x^2}}&{6{x^3}}&{ - 15{x^2}y}&{9{x^4}}&{ - 12{x^3}y}&{21{x^4}y}&{ - 27{x^4}{y^2}} \\
  { - 4xy}&{ - 8{x^2}y}&{20x{y^2}}&{ - 12{x^3}y}&{16{x^2}{y^2}}&{ - 28{x^3}{y^2}}&{36{x^3}{y^3}} \\
  {7{x^2}y}& - & - & - & - & - & - \\
  { - 9{x^2}{y^2}}& - & - & - & - & - & -
\end{array}$
Multiplying $7{x^2}y$ with rest of the terms starting from 2x
$
  7{x^2}y \times 2x = 14{x^3}y \\
  7{x^2}y \times \left( { - 5y} \right) = - 35{x^2}{y^2} \\
  7{x^2}y \times 3{x^2} = 21{x^4}y \\
  7{x^2}y \times \left( { - 4xy} \right) = - 28{x^3}{y^2} \\
  7{x^2}y \times 7{x^2}y = 49{x^4}{y^2} \\
  7{x^2}y \times \left( { - 9{x^2}{y^2}} \right) = - 63{x^4}{y^3} \\
 $
Fill the sixth row with the above values
$\begin{array}{*{20}{c}}
  {\dfrac{{{1^{st}}monomial \to }}{{{2^{nd}}monomial \downarrow }}}&{2x}&{ - 5y}&{3{x^2}}&{ - 4xy}&{7{x^2}y}&{ - 9{x^2}{y^2}} \\
  {2x}&{4{x^2}}&{ - 10xy}&{6{x^3}}&{ - 8{x^2}y}&{14{x^3}y}&{ - 18{x^3}{y^2}} \\
  { - 5y}&{ - 10xy}&{25{y^2}}&{ - 15{x^2}y}&{20x{y^2}}&{ - 35{x^2}{y^2}}&{45{x^2}{y^3}} \\
  {3{x^2}}&{6{x^3}}&{ - 15{x^2}y}&{9{x^4}}&{ - 12{x^3}y}&{21{x^4}y}&{ - 27{x^4}{y^2}} \\
  { - 4xy}&{ - 8{x^2}y}&{20x{y^2}}&{ - 12{x^3}y}&{16{x^2}{y^2}}&{ - 28{x^3}{y^2}}&{36{x^3}{y^3}} \\
  {7{x^2}y}&{14{x^3}y}&{ - 35{x^2}{y^2}}&{21{x^4}y}&{ - 28{x^3}{y^2}}&{49{x^4}{y^2}}&{ - 63{x^4}{y^3}} \\
  { - 9{x^2}{y^2}}& - & - & - & - & - & -
\end{array}$
Multiplying $ - 9{x^2}{y^2}$ with rest of the terms starting from 2x
$
   - 9{x^2}{y^2} \times 2x = - 18{x^3}{y^2} \\
   - 9{x^2}{y^2} \times \left( { - 5y} \right) = 45{x^2}{y^3} \\
   - 9{x^2}{y^2} \times 3{x^2} = - 27{x^4}{y^2} \\
   - 9{x^2}{y^2} \times \left( { - 4xy} \right) = 36{x^3}{y^3} \\
   - 9{x^2}{y^2} \times 7{x^2}y = - 63{x^4}{y^3} \\
   - 9{x^2}{y^2} \times \left( { - 9{x^2}{y^2}} \right) = 81{x^4}{y^4} \\
 $
Fill the seventh row with the above values
$\begin{array}{*{20}{c}}
  {\dfrac{{{1^{st}}monomial \to }}{{{2^{nd}}monomial \downarrow }}}&{2x}&{ - 5y}&{3{x^2}}&{ - 4xy}&{7{x^2}y}&{ - 9{x^2}{y^2}} \\
  {2x}&{4{x^2}}&{ - 10xy}&{6{x^3}}&{ - 8{x^2}y}&{14{x^3}y}&{ - 18{x^3}{y^2}} \\
  { - 5y}&{ - 10xy}&{25{y^2}}&{ - 15{x^2}y}&{20x{y^2}}&{ - 35{x^2}{y^2}}&{45{x^2}{y^3}} \\
  {3{x^2}}&{6{x^3}}&{ - 15{x^2}y}&{9{x^4}}&{ - 12{x^3}y}&{21{x^4}y}&{ - 27{x^4}{y^2}} \\
  { - 4xy}&{ - 8{x^2}y}&{20x{y^2}}&{ - 12{x^3}y}&{16{x^2}{y^2}}&{ - 28{x^3}{y^2}}&{36{x^3}{y^3}} \\
  {7{x^2}y}&{14{x^3}y}&{ - 35{x^2}{y^2}}&{21{x^4}y}&{ - 28{x^3}{y^2}}&{49{x^4}{y^2}}&{ - 63{x^4}{y^3}} \\
  { - 9{x^2}{y^2}}&{ - 18{x^3}{y^2}}&{45{x^2}{y^3}}&{ - 27{x^4}{y^2}}&{36{x^3}{y^3}}&{ - 63{x^4}{y^3}}&{81{x^4}{y^4}}
\end{array}$

Note: A monomial is a polynomial with just one term and the degree of monomials might not be the same all the time.If the monomial is having more than one variables then the degree of the monomial is the sum of the exponents of the variables. The degree of all the constants is zero.