Chapter 8.4, Problem 52PPSE

Solution

To find: The expression equivalent to $\frac{x}{x^2-2x-3}\cdot \frac{2x-6}{x^2-4x+3}$ .

The simplified form of the multiplication is $\frac{x}{x^2-2x-3}\cdot \frac{2x-6}{x^2-4x+3}=\frac{2x}{\left(x-3\right)\left(x+1\right)\left(x-1\right)}$ . So, option $\left(C\right)$

  $\frac{2x}{\left(x-1\right)\left(x+1\right)\left(x-3\right)}$ is correct.

Given:

The expression to evaluate is $\frac{x}{x^2-2x-3}\cdot \frac{2x-6}{x^2-4x+3}$ .

Concept used:

A rational expression $R\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}$ where $P\left(x\right)\text{ and }Q\left(x\right)$ are polynomial and $Q\left(x\right)\ne 0$ , is said to be in its simplest form when its numerator and denominator have no common divisors in it, and in the simplified form of rational expression, the restriction is on the denominator at which the denominator is equal to zero. Further, if the rational expression has common divisors in it, then equate the common divisor equal to zero which will give another restriction that is not evident, but it is needed to prevent the denominator of the original expression from being reduced to zero.

Calculation:

With the help of the concept stated above, the simplified form of the multiplication is evaluated by applying restrictions on the variables wherever required.

Therefore, the rational expression $\frac{x}{x^2-2x-3}\cdot \frac{2x-6}{x^2-4x+3}$ is multiplied as:

  $\begin{array}{c}\frac{x}{x^2-2x-3}\cdot \frac{2x-6}{x^2-4x+3}=\frac{x}{\left(x-3\right)\left(x+1\right)}\cdot \frac{2\left(x-3\right)}{\left(x-3\right)\left(x-1\right)}\mathrm{....}\left(\text{Factorize}\right)\\ =\frac{2x\left(x-3\right)}{\left(x-3\right)\left(x+1\right)\left(x-1\right)\left(x-3\right)}\\ \frac{x}{x^2-2x-3}\cdot \frac{2x-6}{x^2-4x+3}=\frac{2x}{\left(x-1\right)\left(x+1\right)\left(x-3\right)}\mathrm{....}\left(\begin{array}{l}\text{ Cancel the common term where,}\\ \text{ the restriction is }x\ne \left\{-1,1,3\right\}\end{array}\right)\end{array}$

Conclusion:

The simplified form of the multiplication is $\frac{x}{x^2-2x-3}\cdot \frac{2x-6}{x^2-4x+3}=\frac{2x}{\left(x-3\right)\left(x+1\right)\left(x-1\right)}$ for $x\ne \left\{-1,1,3\right\}$ .

Therefore, option $\left(C\right)$ $\frac{2x}{\left(x-1\right)\left(x+1\right)\left(x-3\right)}$ is correct.