Case II.

129. To Reduce a Fraction to an Integral or Mixed Expression.

1. Reduce ${\displaystyle \frac{x^3-1}{x-1}}$ to an integral or mixed expression.

By division, ${\displaystyle \frac{x^3-1}{x-1}}=x^2+x+1$.

2. Reduce ${\displaystyle \frac{x^3-1}{x+1}}$ to an integral or mixed expression.

By division, ${\displaystyle \frac{x^3-1}{x+1}}=x^2-x+1-{\displaystyle \frac{2}{x+1}}$.

reduce a fraction to an integral or mixed expression, therefore,

Divide the numerator by the denominator.

If there is a remainder, this remainder must be written as the numerator of a fraction of which the divisor is the denominator, and this fraction with its proper sign must be annexed to the integral part of the quotient.

Exercise 43.

Reduce to integral or mixed expressions:

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

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

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

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

5. ${\displaystyle \frac{8x^3}{2x+1}}$.

6. ${\displaystyle \frac{5x^3+9x^2+3}{x^2+x-1}}$.

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

8. ${\displaystyle \frac{y^4+y^2{x}^2+x^4}{y^2+yx+x^2}}$.