Before we try to simplify or solve any rational expression or do any mathematical operation with them we need to know what exactly a rational expression is.
A rational expression is a fraction with polynomials as its numerator or denominator. It’s also called a polynomial fraction and all the possible calculations that you can apply on a fraction is also applicable on a rational expression. However the only difference is that since it involves a polynomial function as either its numerator or denominator or both we first need to simplify that to get a real value which in turn will simplify the rational expression.
(x2 – 4) / (x2 + 2x – 8) is a rational expression.
The most important theory applied while simplifying a rational expression is that when dealing with any kind of fraction the denominator can never be equal to zero for the solution to be a real number.
As you can see in the above example, we have polynomial functions both as numerator and denominator. Hence to simplify this expression we need to find the domains of any of the polynomial first.
As explained above, for a rational expression to be true the denominator can never be zero and by this theory we would reduce the quadratic in the denominator to find its domains. So, let’s assume that;
(x2 + 2x – 8) = 0
(x + 4) (x – 2) = 0 [by factoring]
Therefore, either (x + 4) = 0 or (x – 2) = 0
i.e. x = – 4 or x = 2
As actually, (x2 + 2x – 8) ≠ 0, hence x ≠ – 4, 2
So, ‘x’ can be all values except (– 4) and (2) for the above rational expression to be true.
Another method of simplifying rational expressions is by cancelling off the common factors between numerator and denominator as you must have done before for whole number fractions. Similarly we can apply the same theory to polynomial fractions too. For example;
(x2 + 7x + 12) / (x2 + x – 6) = (x + 3) (x + 4) / (x + 3) (x – 2)
Cancelling the common factor (x + 3) from both numerator and denominator we get;
(x + 4) / (x – 2)
Let’s take a look at another tricky example. Simplify the expression: (x – 3) / (3 – x)
Now by cancellation method this looks very close but is not actually same so that the numerator and denominator can be cancelled. However, if we express the denominator in a different way it may look the same on both sides. That is as shown below.
(x – 3) / (3 – x) = (x – 3) / –(x – 3) = –1 (by cancelling (x – 3) from both numerator and denominator of the rational expression. So the answer is
(x – 3) / (3 – x) = –1 when x ≠ 3, because if x = 3 then the denominator will be zero, which is not allowed.
That’s all about rational expressions.