Algebra: Rational Expressions
Simplifying fractions
As fractions of numbers can be simplified, it is sometimes possible to simplify fractions with variables as well.
Simplifying fractions
If the numerator and the denominator of a fraction have a common denominator, you can divide the fraction by this denominator. This can be a number, but also an expression with variables. This process is called simplification.
The result of the simplification is only equal to the original expression if those values of the variables are chosen so that both denominators are not equal to #0#.
Generally one assumes that the numerical values of the variables are chosen such that we stay outside the 'forbidden' areas, which means that the equality of two expressions only applies if the denominators of both fractions are unequal to zero.
Common denominators can be found by factorizing the numerator and the denominator.
Similarly to rational numbers, there are fractions that cannot be simplified, but we will not discuss these here.
If the denominator is equal to #1#, then the result is a polynomial . Compare this with the difference between a rational number and an integer.
An example where the simplified expression does not show the original condition of the denominator is \[\frac{a}{a^2+a}=\frac{1}{a+1}\tiny.\] This equality only applies if #a# is unequal to #0# and #-1#. The condition #a\ne0# can not be derived from the simplified fraction.
For, #{{20\cdot b^2+4\cdot b}\over{\left(b-6\right)\cdot \left(20\cdot b+ 4\right)}} =\frac{ 4 \left( 1+5{b}\right)\cdot {b}}{{4}\left(b-6\right)\left(5{b}+1\right) }={{b}\over{b-6}}#.
If #b = -{{1}\over{5}}#, then the answer is defined, but the original expression is not.
If #b=6#, then both the answer and the original expression are undefined.
Or visit omptest.org if jou are taking an OMPT exam.
