Question

Divide ${x^2} - 7x - 18$ by $x - 9$ .

Answer 1

Hint: First, find the factors of the equation ${x^2} - 7x - 18$ by using the middle term splitting method.
Then, divide the product of the two factors by $x - 9$ , to get the final answer.
Middle term splitting method:
This method is used to split the middle term in such a way that the sum of those is the product of coefficients of first and last terms.

Complete step-by-step answer:
It is asked to divide ${x^2} - 7x - 18$ by $x - 9$ .
We will divide ${x^2} - 7x - 18$ by $x - 9$ by using the method of factorization.
Now, first split the middle term in the equation ${x^2} - 7x - 18$ as $ - 9x + 2x$ .
 $\therefore {x^2} - 7x - 18 = {x^2} - 9x + 2x - 18$
So,
 $
  {x^2} - 9x + 2x - 18 = x\left( {x - 9} \right) + 2\left( {x - 9} \right) \\
   = \left( {x - 9} \right)\left( {x + 2} \right) \\
 $
Now, we will divide $\left( {x - 9} \right)\left( {x + 2} \right)$ by $x - 9$
 $\therefore \dfrac{{\left( {x - 9} \right)\left( {x + 2} \right)}}{{x - 9}} = x + 2$
Thus, on dividing the equation ${x^2} - 7x - 18$ by $x - 9$ , we get $x + 2$ .

Note: Alternate Method:
It is asked to divide ${x^2} - 7x - 18$ by $x - 9$ .
We will divide ${x^2} - 7x - 18$ by $x - 9$ by using the synthetic method.
Firstly, take $x - 9 = 0$ .
Thus, \[x = 9\] .
Now, for the synthetic method of division, we take the coefficients of the equation ${x^2} - 7x - 18$ .
Thus, $9\left| \!{\underline {\,

  \begin{array}{*{20}{c}}
  1&{ - 7}&{ - 18}
\end{array} \\
  \begin{array}{*{20}{c}}
  0&9&{18}
\end{array} \\
  \,}} \right. $
                $\begin{array}{*{20}{c}}
  1&2&0
\end{array}$
After the above step, the first term becomes the coefficient of x, second term becomes the coefficient of constant term and last term is the remainder.
So, we get the equation $x - 2 = 0$ after dividing ${x^2} - 7x - 18$ by $x - 9$ .
Thus, on dividing the equation ${x^2} - 7x - 18$ by $x - 9$ , we get $x + 2$ .