Given an algebraic fraction you know how to simplify it. For example, \[ \begin{align} \frac{x^2-4}{x^2-6x+8} &= \frac{(x-2)(x+2)}{(x-2)(x-4)} \\ &= \frac{x+2}{x-4} \end{align} \ \] Sometimes you will need to go back the other way and break the algebraic fraction up. For example, \[ \begin{align} \frac{x+7}{x+2} &= \frac{x+2+5}{x+2} \\ &= 1 + \frac{5}{x+2} \end{align} \ \] Try this with \[ \begin{align} &\frac{3x-2}{x+5} \\ \end{align} \ \]
(a) Explain why
\[ \begin{align}
\frac{b-a}{y-x} &= \frac{a-b}{x-y}
\end{align} \ \]
(b) Explain why
\[ \begin{align}
\frac{x-d}{y} &= -\frac{d-x}{y}
\end{align} \ \]
How would you use these properties to help simplify an algebraic fraction?
Given an algebraic fraction you know how to reduce it to its lowest form. For example, \[ \begin{align} \frac{8b^3xy^3}{20b^5y^4} &= \frac{2x}{5b^2y} \end{align} \ \] Sometimes you will need to go back the other way and build the algebraic fraction up. For example, find the value of N in \[ \begin{align} \frac{5ab}{a+1} &= \frac{N}{4(a^2-1)(a-2)} \\ \end{align} \ \]