8. Rational expressions*

Rational expressions

If [[$ P $]] and [[$ Q $]] are polynomials (Q ≠ 0), then an expression that is of the form [[$ P : Q $]] can be converted into form:
[[$$ \displaystyle\frac {P} {Q} $$]] This form is called a rational expression.

A rational expression is formed by dividing a polynomial by another polynomial. Rationial expressions express the same phenomenon as fractions, and they are subject to the same calculation rules. However, in rational expressions, the numerator and denominator consist of polynomials instead of individual numeric values. A rational expression is impossible to solve if its denominator is zero, as zero cannot act as the divisor. For the same reason, the rational expression can be defined by any value except those that result in zero becoming the denominator.

Calculation rules for rational expressions

Before rational expressions can be added together or subtracted, they must be coverted into the same form by extending and/or reducing.[[$$ \displaystyle\frac {a} {b} + \displaystyle\frac {c} {d} = \displaystyle\frac {ad + bc} {bd} $$]]
Rational expressions are multiplied so that the numerators are multiplied and the denominators are multiplied. [[$$ \displaystyle\frac {a} {b} \cdot \displaystyle\frac {c} {d} = \displaystyle\frac {ac} {bd} $$]]
Rational expressions can be divided by another fractionial expression by multiplying the dividend by the inverse of the divisor.[[$$ \displaystyle\frac {a} {b} : \displaystyle\frac {c} {d} = \displaystyle\frac {ad} {bc} $$]]

Example 1

a) [[$ \displaystyle\frac {2} {3} + \displaystyle\frac {1} {5} = \displaystyle\frac {^{5)} 2} {\; 3} + \displaystyle\frac {^{3)} 1} {\; 5} = \displaystyle\frac {5 \cdot 2 + 3 \cdot 1} {15} = \displaystyle\frac {13} {15} $]]

b) [[$ \displaystyle\frac {x} {3} - \displaystyle\frac {y} {5} = \displaystyle\frac {^{5)} x} {\;3} - \displaystyle\frac {^{3)} y} {\;5} = \displaystyle\frac {5x - 3y} {15} $]]

c) [[$ \displaystyle\frac {x} {3} \cdot \displaystyle\frac {y} {5} = \displaystyle\frac {x \cdot y} {3 \cdot 5} = \displaystyle\frac {xy} {15} $]]

d) [[$ \displaystyle\frac {x} {3} : \displaystyle\frac {y} {5} = \displaystyle\frac {x} {3} \cdot \displaystyle\frac {5} {y} = \displaystyle\frac {5x} {3y} $]]

Example 2

At which values of [[$ x $]] is the fraction [[$ \displaystyle\frac {3x - 4} {-4x +8} $]] defined?

Solution:

The rational expression is defined at all values that do not result in zero becoming the denominator. Thus, you must find out which value of [[$ x $]] gives the denominator a value of zero.

[[$ \begin{align*} -4x + 8 &= 0 \\ -4x &= -8 \;\;\;\;\; ||: (-4) \\ x &= \displaystyle\frac {-8} {-4} \\ x &= 2 \\ \end{align*} $]]

Answer: The rational expression is defined when [[$ x ≠ 2 $]].