Definition :
An expression is called a rational expression if it can be written in the form P(x) / Q(x) where p(x) and q(x) are polynomials and Q(x) ≠ 0 . A rational expression is the ratio of two polynomials.
The following are examples of rational expressions.
9/x
(2y + 1) / (y^{2} - 4y + 9)
(z^{3} + 5) / (z - 4)
a / (a + 10)
The rational expressions are applied for describing distance-time, modeling multitask problems, to combine workers or machines to complete a job schedule and the usage goes on endlessly.
A rational expression
P(x) / Q(x)
is said to be in its lowest form if
GCD [P(x), Q(x)] = 1
To reduce a rational expression to its lowest form, follow the steps given below.
Step 1 :
Factorize the numerator and the denominator
Step 2 :
If there are common factors in the numerator and denominator, cancel them.
Step 3 :
The resulting expression will be a rational expression in its lowest form.
Example 1 :
Reduce the following rational to its lowest form.
(x - 3) / (x^{2} - 9)
Solution :
= (x - 3) / (x^{2} - 9)
= (x - 3) / (x^{2} - 3^{2})
= (x - 3) / [(x + 3)(x - 3)]
= 1 / (x + 3)
Example 2 :
Reduce the following rational to its lowest form.
(x^{2} - 16) / (x^{2} + 8x + 16)
Solution :
= (x^{2} - 16) / (x^{2} + 8x + 16)
= (x^{2} - 4^{2}) / [(x + 4)(x + 4)]
= [(x + 4)(x - 4)] / [(x + 4)(x + 4)]
= (x - 4) / (x + 4)
A value that makes a rational expression (in its lowest form) undefined is called an Excluded value.
To find excluded value for a given rational expression in its lowest form, say
P(x) / Q(x),
consider the denominator Q(x) = 0.
For example, the rational expression
5 / (x - 10)
is undefined when x = 0.
So, 10 is called an excluded value for
5 / (x - 10)
Example 1 :
Find the excluded values of the following expression (if any).
(x + 10) / 8x
Solution :
The expressions (x + 10) / 8x is undefined when
8x = 0
Then,
x = 0
So, the excluded value is 0.
Example 2 :
Find the excluded values of the following expression (if any).
(7p + 2) / (8p^{2} + 13p + 5)
Solution :
The expressions (7p + 2) / (p^{2} + 5p + 6) is undefined when
p^{2} + 5p + 6 = 0
Then,
(p + 2)(p + 3) = 0
p + 2 = 0 or p + 3 = 0
p = -2 or p = -3
So, the excluded values are -2 and -3.
Example 3 :
Find the excluded values of the following expression (if any).
x / (x^{2} + 1)
Solution :
Here x^{2 }≥ 0 for all values of 'x'.
Then,
x^{2} + 1 > 0 (always)
So, (x^{2} + 1) ≠ 0 for all values of x.
Therefore, there can be no real excluded values for the given rational expression.
You would have studied the concepts of addition, subtraction, multiplication and division of rational numbers. Now let us generalize the above for rational expressions.
To add two or more rational expressions, follow the cases explained below.
Case 1 :
If the denominators of the rational expressions are same, take the denominator once and add the numerators.
Case 2 :
If the denominators of the rational expressions are different, get the same denominator for all the rational expressions. Now, take the denominator once and add the numerators.
Example 1 :
x/(x + 3) + 1/(x + 3)
Solution :
= x/(x + 3) + 1/(x + 3)
= (x + 1) / (x + 3)
Example 2 :
x/(x + 3) + (x + 3)/(x - 3)
Solution :
= [x(x - 3) / (x + 3)(x - 3)] + [(x + 3)/(x + 3)(x - 3)]
= [x(x - 3) + (x + 3)] / [(x + 3)(x - 3)]
= [x(x - 3) + (x + 3)] / [(x + 3)(x - 3)]
= (x^{2} - 3x + x + 3) / (x^{2} - 3^{2})
= (x^{2} - 2x + 3) / (x^{2} - 9)
To subtract a rational expression from another rational expression, follow the cases explained below.
Case 1 :
If the denominators of the two rational expressions are same, take the denominator once and subtract the numerators.
Case 2 :
If the denominators of the two rational expressions are different, get the same denominator for both the rational expressions. Now, take the denominator once and subtract the numerators.
Example 1 :
x/(x - 5) - 5/(x - 5)
Solution :
= x/(x - 5) - 5/(x - 5)
= (x - 5) / (x - 5)
= 1
Example 2 :
x/(x + 2) - 2/(x - 2)
Solution :
= [x(x - 2) / (x + 2)(x - 2)] - [2(x + 2)/(x + 2)(x - 2)]
= [x(x - 2) - 2(x + 2)] / [(x + 2)(x - 2)]
= (x^{2} - 2x - 2x - 4) / (x^{2} - 2^{2})
= (x^{2} - 4x - 4) / (x^{2} - 4)
If p/q and r/s represent two rational expressions where
q ≠ 0 and r ≠ 0,
then their product is
(p/q) x (r/s) = pr / qs
In other words, the product of two rational expression is the product of their numerators divided by the product of their denominators and the resulting expression is then reduced to its lowest form.
Example :
Multiply (x^{3}/9y^{2}) by (27y/x^{5}).
Solution :
= (x^{3}/9y^{2}) x (27y / x^{5})
= (x^{3 }⋅ 27y) / (9y^{2} ⋅ x^{5})
= 3 / x^{2}y
If p/q and r/s represent two rational expressions where
q ≠ 0 and s ≠ 0,
then
(p/q) ÷ (r/s) = (p/q) x (s/r) = ps / qr
Thus division of one rational expression by other is equivalent to the product of first and reciprocal of the second expression. If the resulting expression is not in its lowest form then reduce to its lowest form.
Example :
Divide (14x^{4}/y) by (7x/3y^{4}).
Solution :
= (14x^{4}/y) ÷ (7x/3y^{4})
= (14x^{4}/ y) x (3y^{4}/7x)
= (14x^{4 }⋅ 3y^{4}) / 7xy
= 6x^{3}y^{3}
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