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Rational Expressions

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Rational numbers can form expressions when we join rational numbers with various mathematical operators namely addition, subtraction, multiplication and division. Ratio of two polynomials is called a rational expression.
Rational numbers are the numbers which are written in form of $\frac{a(x)}{b(x)}$, where a(x) and b(x) are integers, and b(x) $\neq$ 0. Always remember all natural, whole numbers, integers including zero are all the members of the family of rational numbers because they all have 1 as the denominator, which is not 0. Rational expressions cannot be defined for numbers that make the denominator zero. A rational function can either be proper or improper. In proper rational function the degree of numerator will be less than the degree of denominator, while in improper the degree of numerator is greater than the denominator. They play an important role in numerical analysis for interpolation and approximation of functions, computer algebra systems, ring theory, wavefunctions etc.,

Define Rational Expressions

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Rational expression is also known as rational function which includes polynomial in numerator and denominator. It should have a variable in the denominator and it looks like a fraction.
It is of the form
f(x) = $\frac{a(x)}{b(x)}$, b(x) cannot be zero.Examples : $\frac{x}{x+9}$, $\frac{x^{2}}{3x^{2}}$, $\frac{8}{x^{2}+7y -6z}$, $\frac{x+4}{x+2}$, $\frac{1}{x^{2} +4}$

Rational Expressions Operation

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Rational expressions operations are explained with the help of examples given below.

Addition
LCM method can be used to solve the rational expression for addition.

Consider: 2x + $\frac{4}{7}$x + $\frac{-3}{2}$x

Find the least common multiple of the given fractions. Numerators of the same fractions are added and simplified.

To simply the given expression, consider the denominators, 1, 7 and 2. So we take the L.C.M, L.C.M of the three numbers is 14.
Multiply numerator and denominator of 2 by 14, numerator and denominator of $\frac{4}{7}$ by 2 and numerator and denominator of $\frac{-3}{2}$ by 7, We get

= $\frac{28}{14}$x + $\frac{8}{14}$ x + $\frac{-21}{14}$x

= $\frac{(28 + 8 -21)(x)}{14}$

= $\frac{15x}{14}$

Subtraction
Subtraction of the rational expression is similar to the addition. Let us study the subtraction of expression with the help of example.

Consider: $\frac{-4}{6}$x - $\frac{-2}{4}$x

In subtraction we first find the lowest common multiple for the given fraction, find difference in the numerator and the lowest common multiple will be in the denominator. Simplify the fraction

L.C.M. of 6 and 4 is 12, So the denominator = 12.
For this we multiply and divide $\frac{-4}{6}$ by 2, and multiply and divide $\frac{-2}{4}$ by 3, We get

= - $\frac{8}{12}$ x - $\frac{-6}{12}$x

= $\frac{(-8 +6)(x)}{12}$

= - $\frac{2x}{12}$

Now further we simplify to get $\frac{-x}{6}$.

Example: $\frac{5}{x^{2} -1}$ - $\frac{1}{x^{2} + 2x + 1}$

Solution: $\frac{5}{(x+1) (x-1)}$ - $\frac{1}{(x+1)^{2}}$

= $\frac{5(x+1)}{(x+1)^{2} (x-1)}$ - $\frac{(x-1)}{(x+1)^{2} (x-1)}$

= $\frac{5(x+1) - (x-1)}{(x+1)^{2} (x-1)}$

= $\frac{5x +5-x+1}{(x+1)^{2}(x-1)}$

= $\frac{4x +6}{(x+1)^{2}(x-1)}$

Multiplication
Multiply the numerators and denominators for the given fractions. Place the product of the numerators over the product of the denominators and then simplify.

Consider:$\frac{5}{x}$ * $\frac{2}{x}$

Solution:
$\frac{5}{x}$ * $\frac{2}{x}$

We get,
= $\frac{10}{x^{2}}$

Simplify: $\frac{x+5}{x}$ $\times$ $\frac{x+2}{x+1}$

Given : $\frac{x+5}{x}$ $\times$ $\frac{x+2}{x+1}$

$\rightarrow$ $\frac{(x+5)(x+2)}{x(x+1)}$

= $\frac{x^{2} + 7x + 10}{x^{2} + x}$

= $\frac{x^{2}(1+\frac{7}{x}+\frac{10}{x^{2}})}{x^{2}(1+\frac{1}{x})}$

= $\frac{1+\frac{7}{x}+\frac{10}{x^{2}}}{1+\frac{1}{x}}$

Division

When a polynomial is divided by a monomial, divide each term in the numerator by the denominator.
Example:
$\frac{x^{3} + 5x^{2} + 6}{2x}$
Solution:
$\frac{x^{3} + 5x^{2} + 6}{2x}$ = $\frac{x^{3}}{2x}$ + $\frac{5x^{2}}{2x}$ + $\frac{6}{2x}$

= $\frac{x^{2}}{2}$ + $\frac{5x}{2}$ + $\frac{3}{x}$

If more than one operator exists in the given expression, then we apply the laws of "PEMDAS" to solve the expressions of rational numbers, where

P : Parentheses
E : Exponents
M : Multiplication
D : Division
A : Addition
A : Subtraction

Steps to Simplify Rational Expressions

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Given below are the steps required to simply rational expressions:
  1. Find the values for which the denominator is equal to zero.
  2. Factor the quadratic to find the values.
  3. Factor the numerator and denominator for the given problem.
  4. The given problem needs to be rewritten as expressions equal to 1.
  5. Avoid writting the values of the variables that result in a denominator of 0.
  6. Simplify

Example 1: Simplify: $\frac{x+4}{x^{2} + 12x + 32 }$ = 0

Solution:
x$^{2}$ + 12x + 32 = 0
(x + 4) (x + 8) = 0
$\Rightarrow$ $\frac{x+4}{(x+4)(x+8)}$

= $\frac{1}{x+8}$

Example 2: Simplify : $\frac{3-x}{x^{2} -9}$
Solution :
$x^{2}$ - 9 = (x + 3) (x - 3)

$\rightarrow$ $\frac{3-x}{(x+3)(x-3)}$

= $\frac{-1}{x+3}$