130. To Reduce a Mixed Expression to a Fraction

130. To Reduce a Mixed Expression to a Fraction.

The process is precisely the same as in Arithmetic. Hence,

Multiply the integral expression by the denominator, to the product add the numerator, and under the result write the denominator.

Reduce to a fraction $a-b-{\displaystyle \frac{a^2-ab-b^2}{a+b}}$.

$$\begin{gathered} a-b-\frac{a^2-ab-b^2}{a+b}\&=\frac{(a-b)(a+b)-(a^2-ab-b^2)}{a+b} \\ \&=\frac{a^2-b^2-a^2+ab+b^2}{a+b} \\ \&=\frac{ab}{a+b}. \end{gathered}$$

The dividing line between the terms of a fraction has the force of a vinculum affecting the numerator. If, therefore, a minus sign precedes the dividing line, as in the preceding Example, and this line is removed, the numerator of the given fraction must be enclosed in a parenthesis preceded by the minus sign, or the sign of every term of the numerator must be changed.

Exercise 44.

Reduce to a fraction:

1. $x-y+{\displaystyle \frac{2xy}{x-y}}$.

2. $x+y-{\displaystyle \frac{2xy}{x+y}}$.

3. $1-{\displaystyle \frac{x-y}{x+y}}$.

4. $a-x-{\displaystyle \frac{a^2+x^2}{a-x}}$.

5. $x+2-{\displaystyle \frac{x^2-4}{x-3}}$.

6. ${\displaystyle \frac{x-3}{x-2}}-2x+1$.

7. ${\displaystyle \frac{x+3}{x+2}}+x^2-x-1$.

8. $2a-1+{\displaystyle \frac{3-4a}{a-3}}$.

9. $1-2a^2-{\displaystyle \frac{a^2-a+2}{a-1}}$.