Question

Carry out the multiplication of expressions in each of the following pairs.
  $ \left( i \right)4p,q + r $
  $ \left( {ii} \right)ab,a - b $
  $ \left( {iii} \right)a + b,7{a^2}{b^2} $
  $ \left( {iv} \right)\left( {{a^2} - 9} \right) \times \left( {4a} \right) $
  $ \left( v \right)pq + qr + rp,0 $

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 answer:
  $ \left( i \right)4p,q + r $
Here there are two terms i.e. \[4p,q + r\] , the first term is a monomial and the second term is a binomial. The terms have the variables p, q and r.
 $ \to 4p \times \left( {q + r} \right) = 4 \times p\left( {q + r} \right) $
Expand the multiplication by distributive property
 $
   = \left( {4 \times p \times q} \right) + \left( {4 \times p \times r} \right) \\
   = 4pq + 4pr \\
  $
The product of \[4p,q + r\] is \[4pq + 4pr\]
  $ \left( {ii} \right)ab,a - b $
Here there are two terms i.e. $ ab,a - b $ , the first term is a monomial and the second term is a binomial. The terms have the variables a, b.
 $
   \to ab \times \left( {a - b} \right) \\
   = a \times b \times \left( {a - b} \right) \\
  $
Expand the multiplication by distributive property
\[
   = \left( {a \times b \times a} \right) - \left( {a \times b \times b} \right) \\
   = \left( {{a^2} \times b} \right) - \left( {a \times {b^2}} \right) \\
   = {a^2}b - a{b^2} \\
 \]
The product of $ ab,a - b $ is \[{a^2}b - a{b^2}\]
  $ \left( {iii} \right)a + b,7{a^2}{b^2} $
Here there are two terms i.e. $ a + b,7{a^2}{b^2} $ , the first term is a binomial and the second term is a monomial. The terms have the variables a, b.
 $ \to \left( {a + b} \right) \times 7{a^2}{b^2} $
Expand the multiplication by distributive property
 $
   = \left( {a \times 7{a^2}{b^2}} \right) + \left( {b \times 7{a^2}{b^2}} \right) \\
   = \left( {7{a^{2 + 1}}{b^2}} \right) + \left( {7{a^2}{b^{2 + 1}}} \right) \\
  \left( {\because {a^m} \times {a^n} = {a^{m + n}}} \right) \\
   = 7{a^3}{b^2} + 7{a^2}{b^3} \\
  $
The product of $ a + b,7{a^2}{b^2} $ is $ 7{a^3}{b^2} + 7{a^2}{b^3} $
  $ \left( {iv} \right)\left( {{a^2} - 9} \right) \times \left( {4a} \right) $
Here there are two terms i.e. $ \left( {{a^2} - 9} \right) \times \left( {4a} \right) $ , the first term is a binomial and the second term is a monomial. The terms have a single variable a.
 $ \to \left( {{a^2} - 9} \right) \times 4a $
Expand the multiplication by distributive property
 $
   = \left( {{a^2} \times 4a} \right) - \left( {9 \times 4a} \right) \\
   = \left( {{a^{2 + 1}} \times 4} \right) - \left( {36a} \right) \\
  \left( {\because {a^m} \times {a^n} = {a^{m + n}}} \right) \\
   = 4{a^3} - 36a \\
  $
The product of $ \left( {{a^2} - 9} \right) \times \left( {4a} \right) $ is $ 4{a^3} - 36a $
  $ \left( v \right)pq + qr + rp,0 $
Here there are two terms i.e. $ pq + qr + rp,0 $ , the first term is a trinomial and the second term is a zero. The first term has three variables p, q and r.
As we know that, any term or any variable or any number multiplied with zero results zero.
So, the product of $ pq + qr + rp,0 $ is $ 0 $

Note: The degree of a monomial is defined as the sum of all the exponents of the variables, including the implicit exponents of 1 for the variables which appear without exponent. The degree of all the constants is zero.