Question

How do you simplify $ \dfrac{{4x - 9}}{{2x - 3}} $ ?

Answer 1

Hint: We can perform the long division to get a simplified version of a rational expression. Also remember the result that $ p(x) = d(x).q(x) + r(x) $ where $ q(x) $ is the quotient and $ r(x) $ when the expression $ p(x) $ is divided by $ d(x) $ .

Complete step-by-step answer:
Here the given expression $ \dfrac{{4x - 9}}{{2x - 3}} $ is a rational expression of the form $ \dfrac{{p(x)}}{{d(x)}} $ where $ p(x){\text{ and }}d(x) $ are polynomials of degree $ 1 $ .
 $ \Rightarrow (4x - 9) = 2(2x - 3) - 3 $
Now, we will put this value of the numerator in the given expression to get,
 $ \dfrac{{2(2x - 3) - 3}}{{2x - 3}} $
 $ \Rightarrow \dfrac{{2(2x - 3)}}{{2x - 3}} - \dfrac{3}{{2x - 3}} $
 $ \Rightarrow 2 - \dfrac{3}{{2x - 3}} $
And this is the required simplified expression of the given rational expression.
So, the correct answer is “ $ 2 - \dfrac{3}{{2x - 3}} $ ”.

Note: To divide rational expressions using long-division the degree of the numerator $ \geqslant $ degree of the denominator polynomial. Cancel out the common factors and remaining will be the required result. Such a simplification of a rational expression is very much useful while finding the limit of the expression and also while differentiating and integrating the expression. As you can see that the simplified expression is easier to integrate than the original expression.
Also, note that the expression $ \dfrac{{4x - 9}}{{2x - 3}} $ is not defined when $ 2x - 3 = 0 \Rightarrow x = \dfrac{3}{2} $ . So one has to be careful regarding this while proceeding for further calculations.