Question

How do you simplify $\dfrac{{2x - 4}}{{{x^2} - 3x + 2}} \times \dfrac{{{x^2} - 4x}}{4}$ ?

Answer 1

Hint: Here we will simplify if the given rational expression is not in its lowest terms, then it can be reduced to its lowest terms by dividing both numerator and denominator by the GCD of the numerator and the denominator.

Complete step by step answer:
If $\dfrac{{p(x)}}{{q(x)}},q(x) \ne 0$ and $\dfrac{{g(x)}}{{h(x)}},h(x) \ne 0$ are two rational expressions, then their product $\dfrac{{p(x)}}{{q(x)}} \times \dfrac{{g(x)}}{{h(x)}}$ is defined as $\dfrac{{p(x) \times g(x)}}{{q(x) \times h(x)}}$ -------------$(A)$
Here the given expression is $\dfrac{{2x - 4}}{{{x^2} - 3x + 2}} \times \dfrac{{{x^2} - 4x}}{4}$
The $p(x) = 2x - 4$ , $q(x) = {x^2} - 3x + 2$ and $g(x) = {x^2} - 4x$ , $h(x) = 4$ then,
Taking the common term in $p(x)$ it becomes $p(x) = 2(x - 2)$--------$(A)$
$q(x)$ is quadratic equation so we can factorize the equation and written in two factors, so we know the factorization process that is,
To factorize ${x^2} - 3x + 2$ one can proceed as follows,
$ \begin{array}{*{20}{c}}
  {\,\,\,\,\,\,\,2} \\
  {\,\,\,\,\,\, \swarrow \,\, - 3\,\,\,\, \searrow }
\end{array} \\
   - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 2 \\
 $ $\because \, - 1 \times - 2 = 2$ and $ - 1 - 2 = - 3$
Therefore the factors are $\left( {x - 1} \right)$ and $\left( {x - 2} \right)$ . then $q(x) = \left( {x - 1} \right)\left( {x - 2} \right)$ --------$(2)$
Now in $g(x) = {x^2} - 4x$ we are going to take the common term $x$ outside from the bracket.
We get, $g(x) = x\left( {x - 4} \right)$----------$(4)$
$h(x) = 4$---------------$(5)$
Substitute the equations $(1),(2),(3),(4){\text{ and (5)}}$ in equation $(A)$ we get,
$\dfrac{{p(x) \times g(x)}}{{q(x) \times h(x)}} = \dfrac{{2x\left( {x - 2} \right)\left( {x - 4} \right)}}{{4\left( {x - 1} \right)\left( {x - 2} \right)}}$
Now divide the term which is common to the denominator and also the numerator, here $2$ is the common term of both numerator and the denominator. And also we can divide the term $\left( {x - 2} \right)$ in numerator and the denominator we get,
$ = \dfrac{{x\left( {x - 4} \right)}}{{2\left( {x - 1} \right)}}$.

Note: Simplification of an algebraic expression can be defined as the process of writing an expression in the most efficient and compact form without affecting the value of the original expression. The process entails collecting like terms, which implies, adding or subtracting terms in an expression. Before you evaluate an algebraic expression, you need to simplify it. This will make all your calculations much easier. Here are the basic steps to follow to simplify an algebraic expression,
$1.$ Remove parentheses by multiplying factors
$2.$ Use exponent rules to remove parentheses in terms with exponents
$3.$ Combine like terms by adding coefficients
$4.$ Combine the constants