Rational Expressions

An expression can express the ratio between two polynomials through the use of fractions. The only rule to these expressions is that the denominator cannot equal zero.

Simplifying

In order to simplify fractions, we first factor the numerator and denominator. After that we cancel out any common factors between the numerator and denominator.

$\underline{Example}$

$\frac{x^2 - 16}{x^2 - 5x + 4}$

$=\frac{(x + 4)(x - 4)}{(x - 1)(x - 4)}$

$=\frac{(x + 4)\bcancel{(x - 4)}}{(x - 1)\bcancel{(x - 4)}}$

$=\frac{(x + 4)}{(x - 1)}$

Adding and Subtracting

To add or subtract fractions, we need their denominators to be equal. We can achieve this by multiplying the numerator and denominator of each fraction with the appropriate expression so that the denominator equals the LCD (least common denominator). After that we just add or subtract the numerators and keep the denominator the same.

$\underline{Example}$

$\frac{12}{2q} - \frac{6}{3p}$ where LCD: $6pq$

$=(\frac{12}{2q} \times \frac{3p}{3p}) - (\frac{6}{3p} \times \frac{2q}{2q})$

$=\frac{36p}{6pq} - \frac{12q}{6pq}$

$=\frac{36p - 12q}{6pq}$

$=\frac{12(3p - q)}{6pq}$

$=\frac{2(3p - q)}{pq}$

Multiplying and Dividing

In order to multiply fractions, we multiply the numerator together and then the denominator together. In order to divide fractions, we multiply the first fraction by the reciprocal of the second fraction. Do note, it can be easier to simplify and do these operations if you factor the fractions first.

$\underline{Example}$

$\frac{3y^2 - 7y - 6}{2y^2 - 3y - 9} \div \frac{y^2 + y - 2}{2y^2 + y - 3}$

$=\frac{(3y + 2)(y - 3)}{(2y + 3)(y - 3)} \div \frac{(y - 1)(y + 2)}{(y - 1)(2y + 3)}$

$=\frac{(3y + 2)\bcancel{(y - 3)}}{(2y + 3)\bcancel{(y - 3)}} \div \frac{\bcancel{(y - 1)}(y + 2)}{\bcancel{(y - 1)}(2y + 3)}$

$=\frac{(3y + 2)}{(2y + 3)} \div \frac{(y + 2)}{(2y + 3)}$

$=\frac{(3y + 2)}{(2y + 3)} \times \frac{(2y + 3)}{(y + 2)}$

$=\frac{(3y + 2)(2y + 3)}{(2y + 3)(y + 2)}$

$=\frac{(3y + 2)\bcancel{(2y + 3)}}{\bcancel{(2y + 3)}(y + 2)}$

$=\frac{3y + 2}{y + 2}$

Complex Fractions

Complex fractions are fractions that contain additional rational expressions in the numerator and/or the denominator. To simplify these fractions, you need to simplify the numerator and denominator into single fractions respectively. You divide the numerator by the denominator and then simplify the resulting fraction.

$\underline{Example}$

$\frac{\frac{6}{y} - \frac{4}{x}}{y}$

$=\frac{\frac{6x}{xy} - \frac{4y}{xy}}{y}$

$=\frac{\frac{6x - 4y}{xy}}{y}$

$=\frac{6x - 4y}{xy} \div \frac{y}{1}$

$=\frac{6x - 4y}{xy} \times \frac{1}{y}$

$=\frac{6x - 4y}{xy^2}$