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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. 

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How to Subtract Rational Expressions?

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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

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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

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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,

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

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

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