Question

Build the rational expression into an equivalent rational expression with the indicated denominator.

Answer 1

Answer: $\frac{m-3}{m+2}\text{ = }\frac{m^2-6m\text{ }+\text{ 9}}{m^2-m-6}$

Explanation:

Given:

$\frac{m-3}{m+2}\text{ = }\frac{unknown}{m^2-m-6}$

To find:

the numerator of the rational expression on the right

To determine the numerator, we need to factorise the denominator:

$\begin{gathered} factors\text{ of -6 whose sum gives -1 are -3 and 2} \\ m^2\text{ - m -6 = m}^2\text{ - 3m + 2m - 6} \\ =\text{ m\lparen m - 3\rparen + 2\lparen m - 3\rparen} \\ =\text{ \lparen m + 2\rparen\lparen m - 3\rparen} \end{gathered}$

$\begin{gathered} \frac{m-3}{m+2}\text{ = }\frac{unknown}{(m\text{ + 2\rparen\lparen m}-3)} \\ from\text{ the above, we see the denominator of the left side was multiplied by \lparen m -3\rparen to get} \\ the\text{ denominator on the right} \\ \\ So\text{ for the expression on the right to be equivalent to that on the left, } \\ \text{we will multiply the numerator on the left by \lparen m - 3\rparen} \end{gathered}$$\begin{gathered} denominator\text{ on the left = m + 2} \\ denominator\text{ on the right = \lparen m + 2\rparen\lparen m - 3\rparen} \\ \\ numerator\text{ on the left = m - 3} \\ numerator\text{ on the right = \lparen m - 3\rparen\lparen m - 3\rparen} \end{gathered}$

Expanding the numerator on the right:

$\begin{gathered} (m\text{ - 3\rparen\lparen m-3\rparen = m\lparen m - 3\rparen - 3\lparen m - 3\rparen} \\ =\text{ m}^2\text{ - 3m - 3m + 9} \\ =\text{ m}^2\text{ - 6m + 9} \end{gathered}$