4.1 Fraction

The quotient of two numbers or algebraic expressions is called a fraction. The quotient is indicated by a bar (—). We write, the dividend above the bar and the divisor below the bar. For example, $\frac{x^2+2}{x-2}$$\frac{x^2 + 2}{x - 2}$ is a fraction with $x - 2 \neq 0$ . If $x - 2 = 0$ , then the fraction is not defined because $x-2=0\Rightarrow x=2$$x - 2 = 0 \Rightarrow x = 2$ which makes the denominator of the fraction zero.

4.1.1 Rational Fraction

An expression of the form $\frac{N(x)}{D(x)}$$\frac{N(x)}{D(x)}$ , where $N(x)$ and $D(x)$ are polynomials in $x$ with real coefficients and $D(x) \neq 0$ , is called a rational fraction.

For example, $\frac{x^2+3}{(x+1{)}^2(x+2)}$$\frac{x^2 + 3}{(x + 1)^2 (x + 2)}$ and $\frac{2x}{(x-1)(x+2)}$$\frac{2x}{(x - 1)(x + 2)}$ are rational fractions.

4.1.2 Proper Fraction:

A rational fraction $\frac{N(x)}{D(x)}$$\frac{N(x)}{D(x)}$ , with $D(x) \neq 0$ is called a proper fraction if degree of the polynomial $N(x)$ in the numerator is less than the degree of the polynomial $D(x)$ in the denominator. For example, $\frac{2}{x+1},\frac{2x-3}{x^2+4}$$\frac{2}{x + 1}, \frac{2x - 3}{x^2 + 4}$ and $\frac{3x^2}{x^3+1}$$\frac{3x^2}{x^3 + 1}$ are proper fractions.