Rational Expressions

Rational expressions show the ratio of two polynomials. It means both the numerator and denominator are polynomials in it. Just like a fraction, it is also a ratio of algebraic expression, which consists of an unknown variable. Although with the help of a calculator we can simplify this kind of expression. This is one of the important topics of Class 8 Maths. We can also perform arithmetic operations such as addition, subtraction and multiplication for these rationals. Just like the fraction, the rationals can be reduced to the lowest possible terms. These polynomial equations could have degree 1 or more than 1.

For example: x+1/x+2, x2+x+1/2x, etc.

To find the root or zero of polynomial expression, we have to put them equal to zero. But to find the zeros of rational function or expressions we have to put only the numerator equal to zero when the expression has been reduced to its lowest terms.

By lowest terms we mean, both the numerator and denominator do not have any common factors. Like in the case of a fraction, say 2/8, it is not in the lowest form. It can be further reduced by taking 2 as a common factor. Thus, ¼ is the lowest form.

Similarly, for example, x2+2x/3x is a rational expression, which is not in its lowest form. But if we take the common factor x from both numerator and denominator, we get (x+2)/3, which is the lowest form of the expression.

Simplification of Rational Expressions

As we know, to add or subtract any two fractions the denominator should be equal for both fractions. The same rule is applicable to rational functions also. Generally, we express the addition and subtraction by the below-given formula:

  • a/c + b/c = (a+b)/c and a/c – b/c = (a – b)/c

Let us take an example of a fraction first. For example, add and subtract ⅔ and ½ .

So first adding both the fractions, we get;

⅔ + ½ = 4 + 3/ 6 = 7/6

Now by subtracting ½ from ⅔, we get;

⅔ – ½ = 4 – 3/6 = ⅙

So you can see here, firstly we have normalised the denominator, by taking the LCM and then performed the operations. Let us take an example for rationals now.

Rational Expressions Example

Solve: 4/x+1 – 1/x + 1

Solution: First we need to solve the denominators of the given expression.

Therefore the least common denominator here will be;

x(x+2)

Now we can multiply with the factors to all three expressions to make the denominator equal.

Hence,

4/x+1 – 1/x + 1/1 = 4x/x(x+2) – x+2/x(x+2) + x(x+2)/x(x+2)

=> 4x-(x+2)+x(x+2)/x(x+2)

After solving the above expression, we get;

=> (4x-x-2+x2+2x)/x(x+2)

=> (x2 + 5x – 2)/x(x+2)

Multiplication and Division of Rational Expressions

Just like adding and subtracting rational functions, we can also multiply and divide them. The general formula is;

  • a/b × c/d = ac/bd
  • a/b ÷ c/d = a/b × d/c = ad/bc

Let us understand with the help of an example.

Example

Simplify: (x2+5x+4) (x+5)/(x2-1)

Solution: By factoring the numerator and denominator, we get;

=>(x+1)(x+4)(x+5)/(x+1)(x-1)

On canceling the common terms, we get;

=>(x+4)(x+5)/x-1