Step-by-Step Guide to Solving Rational Expression Problems
Rational Terms
We have studied that the fraction is made up of two parts numerator and denominator. When the numerator and denominator consists of polynomials such fractions are called as rational expressions.
For example
\[\frac{4x + 2}{3x + 3}\]
\[\frac{5x + 7}{x + 2}\]
We can also perform arithmetic operations such as addition, subtraction and multiplication with the rational terms. Just like the fraction, the rationals can be reduced to the simplified lowest rational terms. These polynomial equations could have more than one degree.
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In this article let us discuss polynomials and rational expressions and simplifying rational expressions.
What is Rational Expressions?
Rational expression can also be said as the ratio of two polynomials expressions.
If p (x) and q (x) are two polynomials, with q (x) ≠ 0, then general form of rational expression is
In rational expression both the numerator and denominator are polynomials.The denominator of rational expression can never be zero. Like algebraic expressions it contains unknown variables. To simplify the rational expression we reduce it to its simplest form.
Simplifying Rational Expressions
The quotient of two polynomial expressions is called a rational expression.
Simplifying the rational expression means to reduce to its lowest terms. Rational expression is in its lowest form if all common factors from the numerator and denominator have been canceled.
1. We first need to factor the polynomials
2. Cancel any common factors from the numerator and denominator of the rational expression
For Example :
3x2 + x
__________
4x2 + x
Factor the numerator and denominator
x( 3x + 1)
= ________
x( 4x + 1)
Now, simplify the expression by canceling the common factor
(3x + 1)
= ______
(4x + 1)
It is the lowest form.
Multiplying Rational Expressions
Multiplication of rational expressions works the same way as multiplication in fractions. We multiply the numerators with the numerator and denominator with the denominator to find the product. Before multiplying, factor the numerators and denominators just as we did when simplifying rational expressions it will make the calculations easier. We can also simplify the product of rational expressions.
Steps to multiply two rational expressions
Factor the numerator and denominator
Multiply numerators.
Multiply denominators.
Simplify
For example :
[\[\frac{(x + 1)}{2x}\]] X [\[\frac{(y + 2)}{(x + 5)}\]]
= \[\frac{(x + 1) X (y + 2)}{2x X (x + 5)}\]
= \[\frac{xy + 2x + y + 2}{2x2 + 10x}\]
Dividing Rational Expressions
Division of rational expressions works the same way as division in fractions. To divide a rational expression by another rational expression, we have to multiply the first expression by the reciprocal of the second expression.
Steps to Divide Two Rational Expression
Write the first rational expression multiplied by the reciprocal of the second.
Factor the numerators and denominators.
Multiply the numerators.
Multiply the denominators.
Simplify.
For Example
(2x2 + x - 6/x2 - 1) (x2 - 4 / x2 + 2x -1)
Solution:
Write the first rational expression multiplied by the reciprocal of the second.
(2x2 + x - 6/x2 - 1) x ( x2 + 2x - 1/ x2 - 4)
Factor the numerators and denominators.
[(2x -3)( x + 2)/ (x + 1)( x- 1) ] x [ (x+1)2/(x + 2)( x - 2)]
Cancel the common factors
[(2x + 3)(x + 1) / ( x- 1)(x -2)]
Adding and Subtracting Rational Expressions
Adding and subtracting rational expressions works just like adding and subtracting general fractions. To add fractions, we need to find a common denominator.
We have to rewrite the fractions with a common denominator before we are able to add. We must do the same thing when subtracting rational expressions.
The general rule for adding or subtracting the rational expression is
\[\frac{a}{b}\] + \[\frac{c}{d}\] = \[\frac{a \times d + b \times c}{b \times d}\]
The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the least common multiple that the denominators have in common.
Steps to Add or Subtract the Rational Expressions are
Factor the numerator and denominator.
Find the LCD of the expressions.
Multiply the expressions by LCD that changes the denominators to the LCD.
Add or subtract the numerators.
Simplify.
For example:
Add 5/x + 2/y
Solution: first let us find the LCD i.e xy
Now taking xy as the denominator and dividing it by the individual denominator and multiplying by each fraction.
(5/x) x (y/y) + (2/y) x (x/x)
= 5y/xy + 2x/xy
= (5y + 2x) / xy
Solved Examples
Multiply ((x + 1)2 / 8x2 ) x ( 4x / (x + 1))
Solution:
=[(x + 1)( x + 1) / 4 x 2 x x] x [ 4 x / (x + 1)]
Cancel the common factors
=( x + 1) / 2x
Divide 10 / (3x + 1) (2 x + 7) 8 / (3x + 1) (x + 2)
Solution:
Take the reciprocal of second rational expression and multiply
=[2 / ( 3x + 1)( 2x + 7)] x [ ( 3x + 1) ( x+ 2) /8)]
cancel the common factors
= 5 (x + 2) / 4( 2x + 7)
Quiz Time
Solve the following
5x2 / 7x 10x2 / 14x
[(x - 1)(x2 + 2xy + y2) / 3x + 2 ] x [ (x + 1)/ (x + 1)2
FAQs on Rational Expression: Concepts, Operations & Examples
1. What is a rational expression? Give an example.
A rational expression is a fraction in which both the numerator and the denominator are polynomials. It can be written in the form P(x)/Q(x), where P(x) and Q(x) are polynomials, and the denominator Q(x) cannot be equal to zero. For example, (x² + 5x + 6) / (x + 2) is a rational expression because both the numerator and denominator are polynomials.
2. How can you identify if an algebraic expression is rational?
To identify a rational expression, you must check two conditions: 1) The expression must be a fraction. 2) Both the numerator and the denominator of the fraction must be polynomials. An expression is not rational if it contains roots of variables (like √x) or variables as exponents (like 2^x) in the numerator or denominator.
3. What are the four basic operations performed on rational expressions?
The four basic arithmetic operations on rational expressions are the same as those for numerical fractions:
Addition and Subtraction: Find a common denominator (the LCD), rewrite each expression with this denominator, and then add or subtract the numerators.
Multiplication: Multiply the numerators together and the denominators together. It's best to factor all polynomials first to simplify by cancelling common factors.
Division: Invert the second rational expression (the divisor) and then multiply it by the first one.
4. Why is finding the domain crucial when working with rational expressions?
Finding the domain is crucial because division by zero is undefined in mathematics. A rational expression has a variable in its denominator. The domain consists of all the real values of the variable for which the expression is defined. We must identify and exclude any values that would make the denominator equal to zero. This prevents mathematical errors and is the first step before simplifying or solving equations involving the expression.
5. How is simplifying a rational expression related to simplifying a numerical fraction?
The process is conceptually identical. To simplify a numerical fraction like 12/18, we find the greatest common factor (6) and cancel it from the numerator and denominator (12/18 = (2×6)/(3×6) = 2/3). Similarly, to simplify a rational expression, we factor the numerator and denominator polynomials and then cancel out any common polynomial factors. The core principle of cancelling common factors to reduce an expression to its simplest form remains the same.
6. What is the key difference between a rational expression and a rational function?
The key difference lies in their purpose. A rational expression is a single algebraic object, like (3x + 2)/(x - 1). A rational function, on the other hand, uses this expression to define a relationship between an input 'x' and an output 'y' or 'f(x)'. It is written as an equation, such as f(x) = (3x + 2)/(x - 1). In simple terms, the expression is the 'rule', while the function is the complete input-output machine defined by that rule.
7. Is a polynomial also a rational expression? Explain why.
Yes, every polynomial is a special case of a rational expression. Any polynomial, let's say P(x), can be written as a fraction by placing it over the number 1, resulting in P(x)/1. Since P(x) is a polynomial and 1 is also a constant polynomial (and is not zero), this form perfectly satisfies the definition of a rational expression. For example, the polynomial x² + 4 can be written as (x² + 4)/1.
