Question

Evaluate the following expressions.
(i) $ \left( {10x - 25} \right) \div 5 $
(ii) $ 10y\left( {6y + 21} \right) \div 5\left( {2y + 7} \right) $
(iii) $ 96abc\left( {3a - 12} \right)\left( {5b - 30} \right) \div 144\left( {a - 4} \right)\left( {b - 6} \right) $

Answer 1

Hint: When the dividend has two or more terms which are joined by multiplication then multiply all the terms with one another and convert the dividend into a single term and take out what is similar in the elements of the dividend. Do the same with the divisor too. If the divisor is a factor of the dividend, then the remainder when their division happens will be zero.

Complete step-by-step answer:
We are given three expressions. Each expression has a dividend and a divisor. We have to find their divisions.
(i) $ \left( {10x - 25} \right) \div 5 $
The dividend is $ \left( {10x - 25} \right) $ and the divisor is the constant 5.
On factoring the dividend, we get
 $
  \left( {10x - 25} \right) = \left( {5 \times 2} \right)x - \left( {5 \times 5} \right) \\
\Rightarrow \left( {10x - 25} \right) = 5\left( {2x - 5} \right) \\
  $
As we can see, the dividend has a factor 5 which is our divisor, so the remainder will be zero
Then the quotient is $ \dfrac{{\left( {10x - 25} \right)}}{5} = \dfrac{{5\left( {2x - 5} \right)}}{5} = 2x - 5 $

(ii) $ 10y\left( {6y + 21} \right) \div 5\left( {2y + 7} \right) $
Expand the dividend first.
 $
\Rightarrow 10y\left( {6y + 21} \right) \\
   = \left( {10y \times 6y} \right) + \left( {10y + 21} \right) \\
   = 60{y^2} + 210y \\
   = 6y\left( {10y + 35} \right) \\
  $
As we can see, the dividend has a factor $ \left( {10y + 35} \right) $ which is our divisor, so the remainder will be zero.
The degree of the dividend is 2 and the degree of the divisor is 1, so the quotient will have a degree $ 2 - 1 = 1 $
On dividing the dividend with the divisor the quotient will be
 $\Rightarrow \dfrac{{10y\left( {6y + 21} \right)}}{{5\left( {2y + 7} \right)}} = \dfrac{{60{y^2} + 210y}}{{10y + 35}} = \dfrac{{6y\left( {10y + 35} \right)}}{{10y + 35}} = 6y $

(iii) $ 96abc\left( {3a - 12} \right)\left( {5b - 30} \right) \div 144\left( {a - 4} \right)\left( {b - 6} \right) $
Factorize the dividend
 $
\Rightarrow 96abc\left( {3a - 12} \right)\left( {5b - 30} \right) = 96abc\left[ {3\left( {a - 4} \right)} \right]\left[ {5\left( {b - 6} \right)} \right] \\
   = 96 \times 3 \times 5\left( {abc} \right)\left( {a - 4} \right)\left( {b - 6} \right) \\
   = 144 \times 10\left( {abc} \right)\left( {a - 4} \right)\left( {b - 6} \right) \\
  $
As we can see, the dividend has a factor $ 144\left( {a - 4} \right)\left( {b - 6} \right) $ which is our divisor, so the remainder will be zero.
Then the quotient is
 $\Rightarrow \dfrac{{96abc\left( {3a - 12} \right)\left( {5b - 30} \right)}}{{144\left( {a - 4} \right)\left( {b - 6} \right)}} = \dfrac{{144 \times 10\left( {abc} \right)\left( {a - 4} \right)\left( {b - 6} \right)}}{{144\left( {a - 4} \right)\left( {b - 6} \right)}} = \dfrac{{144\left( {a - 4} \right)\left( {b - 6} \right) \times 10\left( {abc} \right)}}{{144\left( {a - 4} \right)\left( {b - 6} \right)}} = 10abc $

Note: When the dividend is in terms of a variable and the divisor is also in terms of the same variable and the divisor is a factor of dividend, then the quotient will be a constant only when the degree of dividend and divisor is same. When their degrees differ then the degree of the quotient will be the difference of their degrees.