Question

How do you find the quotient of $ \dfrac{{{x^2} + 5x + 6}}{{x - 4}} $ ?

Answer 1

Hint: We are given a fraction whose numerator and denominator are polynomials. When a polynomial equation of degree n is divided by another polynomial of degree m such that $ m \leqslant n $ , then the degree of the quotient is $ n - m $ and the degree of the remainder is $ m - 1 $ . We will suppose the quotient to be a polynomial of degree 1 and the remainder to be a constant As we have to find the quotient of the given fraction so we will use the division algorithm, and then get the value of the quotient.

Complete step by step solution:
We have to find the quotient of $ \dfrac{{{x^2} + 5x + 6}}{{x - 4}} $
We know that $ a = bq + r $
We have $ a = {x^2} + 5x + 6 $ and $ b = x - 4 $
As the degree of the numerator is 2 and the degree of the denominator is 1, so the degree of the quotient will be 1 and the degree of the remainder will be 0.
So, let $ q = ax + b $ and $ r = r $
 $
   \Rightarrow {x^2} + 5x + 6 = (x - 4)(ax + b) + r \\
   \Rightarrow {x^2} + 5x + 6 = a{x^2} + bx - 4ax - 4b + r \\
   \Rightarrow {x^2} + 5x + 6 = a{x^2} + (b - 4a)x - 4b + r \;
  $
On comparing the two sides of the equation, we get –
 $
  a = 1,\,b - 4a = 5,\, - 4b + r = 6 \\
  Now,\,b - 4a = 5 \\
   \Rightarrow b - 4(1) = 5 \\
   \Rightarrow b = 9 \\
  And\, - 4b + r = 6 \\
   \Rightarrow - 4(9) + r = 6 \\
   \Rightarrow r = 42 \;
  $
So,
 $
  q = 1x + 9 \\
   \Rightarrow q = x + 9 \;
  $
Hence the quotient of $ \dfrac{{{x^2} + 5x + 6}}{{x - 4}} $ is $ x + 9 $ .
So, the correct answer is “ $ x + 9 $ ”.

Note: In this question, we are given two polynomial equations in the numerator and the denominator, the degree of the equation in the numerator is 2 and the degree of that in the denominator is 1. On the factorization of the equation in the numerator, we see that –
 $
  {x^2} + 5x + 6 = {x^2} + 3x + 2x + 6 \\
   \Rightarrow {x^2} + 5x + 6 = x(x + 3) + 2(x + 3) \\
   \Rightarrow {x^2} + 5x + 6 = (x + 3)(x + 2) \;
  $
We get –
 $ \dfrac{{{x^2} + 5x + 6}}{{x - 4}} = \dfrac{{(x + 3)(x + 2)}}{{(x - 4)}} $
We see that the numerator and the denominator do not have any common factor, that’s why we solve the question by applying the division algorithm.