Top

# Subtracting Rational Expressions

A rational expression is a fraction whose denominator and numerator are polynomials. In this section we will study how to use the rules for subtract rational fractions. General expression for the rational expression is, $\frac{P(x)}{Q(x)}$ with $Q(x)\ \neq\ 0$, where, $P(x)$ and $Q(x)$ are polynomials.

Subtracting rational expression is nothing but the process of subtracting the two rational expression polynomials.

 Related Calculators Subtract Rational Expressions Calculator Calculator for Rational Expressions Divide Rational Expressions Calculator Lcd of Rational Expressions Calculator

## How to Subtract Rational Expressions?

Some of the conditions for subtracting rational expressions are given as follows,

• When denominators are same: To subtract the rational expression, the denominator must be same. If yes, Subtract their numerators and write the difference over the common denominator.
• When denominators are different: If the rational expressions denominators are not same, make them same by multiplying and dividing by the same polynomials. And Repeat above step.

## Subtracting Rational Expressions with Different Denominators

Consider two rational numbers $\frac{p(x)}{q(x)}$ & $\frac{r(x)}{s(x)}$ .

Step 1: L.C.M. of “$q(x)$” and “$s(x)$” is $q(x)s(x)$.

Step 2: After multiply the LCM, we have new fractions are

$\frac{p(x)s(x)}{q(x)s(x)}$ & $\frac{r(x)q(x)}{q(x)s(x)}$

Step 4: Now, we have like denominators. Subtract numerator of second fraction from first.

$\frac{p(x)s(x)-r(x)q(x)}{q(x)s(x)}$

Step 5: Always try to convert it in the smallest from. If possible.

## Subtracting Rational Expressions Examples

Let us see some example problems with the same denominator and various denominators in detail and the example problems for subtracting rational expressions to get a better understand of counting fraction.

### Solved Examples

Question 1: Solve the polynomial expression $\frac{2}{x-1}$ - $\frac{1}{x-1}$
Solution:
Given rational expression is $\frac{2}{x-1}$ - $\frac{1}{x-1}$

In above expression, the denominators are common. Hence, the numerator can be subtracted directly.

$\frac{2}{x-1}$ - $\frac{1}{x-1}$ = $\frac{2 - 1}{x-1}$

= $\frac{1}{x-1}$

Therefore, the simplified rational expression is $\frac{1}{x-1}$.

Question 2: Solve the polynomial expression $\frac{2}{x-1}$ - $\frac{1}{(x+1)(x-1)}$
Solution:

Given rational expression is $\frac{2}{x-1}$ - $\frac{1}{(x+1)(x-1)}$

In above expression, the denominators are not common. Hence, to make it common, a polynomial is multiplied and divided.

$\frac{2(x + 1)}{(x-1)(x + 1)}$ - $\frac{1}{(x+1)(x-1)}$

Now, the denominators are common. So, subtract them directly by keeping the denominator as it is.

= $\frac{2x + 2 - 1}{(x-1)(x + 1)}$

Multiply the value inside for further simplification,

= $\frac{2x + 1}{(x-1)(x + 1)}$

=
$\frac{2x + 1}{x^2 - 1}$

 Related Topics Math Help Online Online Math Tutor
*AP and SAT are registered trademarks of the College Board.