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.
Learn: Algebra
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, x^{2}+x+1/2x, etc.
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What is a Rational Expression?
You must have learned about rational numbers, which are expressed in the form of p/q. Rational expressions, on the other hand, are the ratio of two polynomials.
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, x^{2}+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
Below are the important points to be remembered while performing the basic arithmetic operations such as addition, subtraction, multiplication and division on rational expressions:
While reducing rational expressions to the reduced form, the primary step is to factor the numerator and the denominator. After that, reduce or cancel out the common factors of the expressions.
Addition and subtraction: Rational expressions having the same (or like/ common) denominator, keep the denominator as it is and then, add or subtract the numerators. Reduce the remaining expression if possible.
If the rational expressions have different denominators, then first LCM. Now, change each rational expression to the equivalent one by making the denominator exactly the same. Finally add or subtract like terms. Reduce the remaining expression if possible.
Multiplication: Factor the numerators and denominators that are polynomials (if exist any); then, reduce wherever possible. Multiplying the remaining numerators and denominators separately together will result in reduced form.
Division: Invert the denominator (or divisor) and multiply it with the first rational expression (i.e. dividend or numerator) because reducing can be done easily only after converting the division into multiplication similar in the case of dividing fractions. Further simplification is similar to multiplication as explained above.
Addition and Subtraction 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 performing the operations. Let us take an example for rational expressions now.
Now, look at the example given below to understand how to add and subtract rational expressions.
Question 1:
Add 5/(x + 1) and (x + 2)/(x + 1).
Solution:
Given rational expressions are: 5/(x + 1) and (x + 2)/(x + 1)
So,
[5/(x + 1)] + [(x + 2)/(x + 1)]
Here, the denominators are the same.
= [5 + (x + 2)]/ (x + 1)
= (x + 7)/(x + 1)
Question 2:
Subtract 1/(x^{2} – 4) from 7/(x + 2).
Solution:
Given rational expressions are: 1/(x^{2} – 4) from 7/(x + 2)
Now,
[7/(x + 2)] – [1/(x^{2} – 4)]
= [7/(x + 2)] – [1/(x + 2)(x – 2)] {since x^{2} – 4 = x^{2} – 2^{2} = (x + 2)(x – 2)}
Here, the denominators are different so we need to take the LCM of denominators and perform the cross multiplication.
= [7(x – 2) – 1]/ [(x + 2)(x – 2]
= (7x – 14 – 1)/ (x^{2} – 2^{2})
= (7x – 15)/(x^{2} – 4)
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
Consider the below example to understand the multiplication of two rational expressions.
Question:
Multiply \(\frac{x^2-4}{x^2-3x + 2}\) and \(\frac{x^2-5x -14}{x^2-14x+49}\).
Solution:
We can write the given operation as:
\(\frac{x^2-4}{x^2-3x + 2}\times \frac{x^2-5x -14}{x^2-14x+49}\)
Now, we need to factorize the expressions in numerator and denominator.
x^{2} – 4 = x^{2} – 22 = (x – 2)(x + 2)
x^{2} – 3x + 2 = x^{2} – x – 2x + 2 = x(x – 1) – 2(x – 1) = (x – 2)(x – 1)
x^{2} – 5x – 14 = x^{2} + 2x – 7x – 14 = x(x + 2) – 7(x + 2) = (x – 7)(x + 2)
x^{2} – 14x + 49 = x^{2} – 7x – 7x + 49 = x(x – 7) – 7(x – 7) = (x – 7)(x – 7)
So,
\(\frac{x^2-4}{x^2-3x + 2}\times \frac{x^2-5x -14}{x^2-14x+49} = \frac{(x-2)(x+2)}{(x – 2)(x-1)}\times \frac{(x-7)(x+2)}{(x-7)(x-7)}\)
Therefore, the resultant rational expression is: \(\frac{(x+2)^2}{(x-1)(x-7)}\)
Note: Division of rational expressions can be performed by converting the division into multiplication.
Let us understand these operations with the help of examples given below.
Rational Expressions Examples
1.) 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+x^{2}+2x)/x(x+2)
=> (x^{2} + 5x – 2)/x(x+2)
2.) Simplify: (x^{2}+5x+4) (x+5)/(x^{2}-1)
Solution: By factoring the numerator and denominator, we get;
=>(x+1)(x+4)(x+5)/(x+1)(x-1)
On cancelling the common terms, we get;
Practice Problems
- Add (x^{2} – 9)/(x^{2} + 5x + 6) and (x – 3)/(x + 2).
- Subtract (x – 3)/(x + 4) from 6/(x + 3).
- Divide (2x/x^{2} – 9) by 2/(x + 3).