Question

Factorise the expression and divide them as directed.
 $ \left( {{y^2} + 7y + 10} \right) \div \left( {y + 5} \right) $
 $
  A.\,\,y - 2 \\
  B.\,\,y + 1 \\
  C.\,\,y + 2 \\
  D.\,\,y - 1 \\
  $

Answer 1

Hint: To divide given expression we first make factors of given quadratic equation by middle term splitting method and then cancelling common term or factor from numerator and denominator to get value of given expression and hence required solution of given problem.

Complete step-by-step answer:
Given, equation is $ \left( {{y^2} + 7y + 10} \right) \div \left( {y + 5} \right) $
Here, in above we see that there are two terms.
In numerator we have a quadratic equation and in the denominator we have a linear term.
To divide a given expression we first factorise the quadratic equation given in the numerator by the middle term splitting method.
Given quadratic equation is $ {y^2} + 7y + 10 $
Writing $ 7y\,\,as\,\,sum\,\,of\,\,5y\,\,and\,\,\,2y $ . We have
 $
  {y^2} + 5y + 2y + 10 \\
  taking\,\,common\,\,from\,\,above\,\,equation.\,\,We\,\,have, \\
  y\left( {y + 5} \right) + 2\left( {y + 5} \right) \\
   \Rightarrow \left( {y + 5} \right)\left( {y + 2} \right) \\
  $
Now, using factors of given quadratic equation in given equation. We have,
 $
  \dfrac{{{y^2} + 7y + 10}}{{y + 5}} \\
  or\,\,we\,\,can\,\,write\,\,above\,\,equation.\,\,we\,\,have, \\
  \dfrac{{\left( {y + 2} \right)\left( {y + 5} \right)}}{{\left( {y + 5} \right)}} \\
  Canceling\,\,same\,\,terms\,\,in\,\,above\,\,equation. \\
   = y + 2 \;
  $
Therefore, from above we see that the value of the given expression $ \left( {{y^2} + 7y + 10} \right) \div \left( {y + 5} \right) $ is $ y + 2 $ .
So, the correct answer is “Option C”.

Note: The quadratic equation only contains powers of x that are non-negative integers, and therefore it is a polynomial equation. In particular, it is a second-degree polynomial equation, since the greatest power is two.