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Rational expressions are fractions that contain variables in their denominators and often numerators, too. Addition, subtraction, multiplication, and division fractions are four operations performed on rational expressions.
- Complex rational expressions can be simplified by basic mathematical operations.
- Rational expressions represent the division of one polynomial expression by another, too.
- It is a form of algebraic expressions that contain unknown variables.
- If you want an explanation in simple terms, we can also call rational expression an algebraic expression.
- You can also represent this expression in terms of rational fractions.
- The application of expression is seen in the field of finance and investment.
- Fractions are in the form of “p/q”,where q ≠ 0.
- To take an example \(\frac{x^2}{x+3}\) represents the division x2 by x+3 for which a quotient and a remainder can be found.
Key Terms: Rational Expressions, Algebraic Expressions, Rational Terms, Monomial, Binomial, Polynomials, Numerator, Denominator, Fractions, Addition, Subtraction, Multiplication, Division, Arithmetic Operations
What is a Rational Expressions
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Rational expressions are expressions made from the variables and the constants. A variable can take any value. The value of an expression varies with the value you have chosen for variables.
- The denominator cannot be zero (and when the denominator is 0, it becomes undefined)
- Expressions having one, two or three terms are called monomial, binomial and trinomial respectively.
- Any expression made up of one or more terms is called a polynomial.
- We can easily obtain a monomial by multiplying it with another monomial.
- The numerical of a term is called the coefficient.
Frequently used identities:
(a + b)2 = a2 + b2 + 2ab
(a – b)2 = a2 + b2 – 2ab
(a + b) (a – b) =a2 – b2
(x + a) (x + b) = x2 + x (a + b) + ab
Example of What is a Rational Expression?Example: Multiply (x + 5) with (4x - 3) Ans: Multiplication of (x + 5) and (4x - 3) = x (4x - 3) + 5 (4x - 3) = 4x2 - 3x + 20x - 15 = 4x2 + 17x - 15 |
Rational Expressions
Read More:
| Chapter Related Concepts | ||
|---|---|---|
| Cosine Rule | Operations on Rational Numbers | Definite Integrals |
| Real Numbers Formula | Degree of polynomial | Number Systems |
Rational Terms
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Some important rational terms are as follows:
Fraction
Fraction is made up of numerator and denominator. When the numerator and denominator are made of polynomials then such fractions are called rational expressions.
- When we divide the slice of watermelon into four equal parts then each part is equal 1/4th of the whole.
Example of FractionExample: For example \(\frac{4x+3}{2x-1}\) |
Arithmetic
Arithmetic operations such as addition, subtraction and multiplication can be operated with rational terms. Like the fraction, the rational expression can be reduced to the simplified lowest rational terms.
- These polynomial equations can have more than one exponent or power.
- You can see the most common example of arithmetic in your home, which includes stacking of cups.
Example of ArithmeticExample: (x2 – 3x + 2) / (x2 + 2x + 1) |
Simplification of Rational Expressions
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Simplification of rational expressions is a process of reducing the arithmetic expression to the lowest form. The process is similar to that of simplifying fractions.
- A fraction is considered simplified when it has no factor other than one.
- Simplification is done by removing the common factors of the expressions.
Steps to simplify Rational Expressions
Steps to simplify:
- Step 1: Factorize both the denominator and numerator of the rational expression in standard form.
- Step 2: Reduce the expression by canceling common factors in the numerator and denominator.
- Step 3: Finally, it would be best if you rewrote the remaining factors in the numerator and denominator.
Example of Simplication of Rational ExpressionsExample: Simplify (x2 – 4) / (x2 + 4x + 4) Ans: Factor both the numerator and denominator: = (x + 2) (x – 2) / (x + 2) (x + 2) Common factors in the numerator and denominator are canceled out to get. = (x – 2) / (x + 2) |
Simplification of Rational Expressions
Addition and Subtraction of Rational Expressions
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For addition and subtraction of rational expressions, the denominator of both fractions must be equal. If the denominators are not equal, then simplify until they become equal.
- Adding and subtracting rational expressions works like adding and subtracting general fractions.
- To add fractions, first, you must determine the common denominator.
- The fractions with a common denominator are rewritten before adding.
- You can follow the same steps for subtracting rational expressions.
- The LCD (least common denominator) is the least common multiple that the denominators have in common.
- The usual rule for adding or subtracting the rational expression is
\(\frac{a}{b} + \frac{c}{d} = \frac{a \times d + b \times c}{b \times d}\)
Steps for Addition and Subtraction of the Rational Expressions
The steps for additon and subtraction are as follows:
- Step 1: First, numerator and denominator are factorized.
- Step 2: Find the LCD of the expressions.
- Step 3: Multiply the expressions by LCD that changes the denominators to the LCD.
- Step 4: Finally, you must perform addition or subtraction of the numerators.
Example of Addition and Subtraction of Rational ExpressionsExample: Add 5/x + 2/y Ans: Find the LCD i.e xy Then by taking xy as the denominator and dividing it by the individual denominator and multiplying by each fraction. (5/x) × (y/y) + (2/y) × (x/x) = 5y/xy + 2x/xy = (5y + 2x) / xy |
Read More:
| Class 10 Maths Related Concept | ||
|---|---|---|
| Root 2 is an irrational number | Descending Order | Standard Algebraic Identities |
| Natural numbers | Number Lines | Rational Numbers on Number Line |
Multiplication of Rational Expressions
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Multiplication of rational expressions works the same method as multiplication in fractions. In this multiplication of the numerators with the numerator and denominator with the denominator is done to get the product.
- Before multiplying, you must factorize the numerators and denominators, as it makes the calculations easier.
- Then the product of rational expressions is simplified.
a/b × c/d = ac/bd
Steps for Multiplication of Rational Expressions
Steps to multiply two rational expressions
- Factor the numerator and denominator
- First, you must multiply the numerators.
- Lastly, multiply denominators.
Example of Multiplication of the Rational ExpressionsExample: Solve: 4/(x+1) – 1/x + 1 Ans: Solve the denominators of the given expression. Therefore, the least common denominator here will be x(x+2) Now multiply with the factors to all three expressions to make the denominator equal. Hence, 4/(x+2) - 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: = (4x - x - 2 + x2+2x)/x(x+2) = (x2 + 5x - 2)/x(x+2) |
Division of Rational Expressions
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Division of rational expressions is done in the same way division is done in fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second expression.
- The procedure is similar to that of multiplication of rational expressions.
a/b ÷ c/d = a/b × d/c = ad/bc
Steps for Division of Rational Expressions
Steps to divide two rational expressions:
- The first rational expression is multiplied by the reciprocal of the second.
- Factorize the numerators and denominators.
- First, you must multiply the numerators.
- Lastly, multiply denominators.
Example of Divsion of Rational ExpressionsExample: (2x2 + x - 6/x2 - 1) / (x2 - 4 / x2 + 2x -1) Ans: The first rational expression is multiplied by the reciprocal of the second. (2x2 + x - 6/x2 - 1) × (x2 + 2x - 1/ x2 - 4) Factorize the numerators and denominators. (2x−3) (x+2)/(x+1) (x−1) (2x−3) (x+2)/(x+1) (x−1) × [ (x+1)2/ (x + 2) (x - 2)] Canceling the common factors (2x+3) (x+1)/(x−1) (x−2) |
Things to Remember
- Rational expressions is ration of two polynomial equation where numerator and denominator to their lowest form.
- In such expressions, numerator and denominator does not have common value.
- Monomials are expressions with only one term.
- Polynomials are expressions that contain more than two terms with non-zero coefficients and non-negative integral exponents.
- Calculation of the exponent or power of each algebraic factor is done by calculating the algebraic sum of the algebraic factor's exponents in both the monomials.
Sample questions
Ques: Add p (p - q), q (q - r) with r (r - p). (3 marks)
Ans: According to the questions, p (p - q) + q (q - r) + r (r - p)
= p × p - p × q + q × q - q × r + r × r - r × p
= p2 - pq + q2 - qr + r2 - pr
= p2 + q2 + r2 - pq - qr - pr
Ques: Multiply (2x + 5) with (4x - 3). (3 marks)
Ans: Multiplication of (2x + 5) and (4x - 3)
= 2x (4x - 3) + 5 (4x - 3)
= 8x2 - 6x + 20x - 15
= 8x2 + 14x - 15
Ques: Find the product of (2pq + 3q2) and (3pq – 2q2). (3 marks)
Ans: (2pq + 3q2) × (3pq – 2q2)
= 2pq (3pq - 2q2) + 3q2(3pq - 2q2)
= 6p2q2 - 4pq3 + 9pq3 - 6q4
= 6p2q2 + 5pq3 - 6q4
Ques: Simplify (xy + 3z)2. (2 marks)
Ans: (xy + 3z)2
= (xy)2 + 2(xy)(3z) + (3z)2 {Using: (a + b)2 = a2 + 2ab + b2}
= x2y2 + 6xyz + 9z2
Ques: Show that (3x + 7)2 - 84x = (3x - 7)2 .(4 marks)
Ans: LHS = (3x + 7)2 - 84x
= (9x2 + 42x + 49) - 84x
= 9x2 - 42x + 49
= (3x)2 - 2(3x)(7) + (7)2
= (3x - 7)2 = RHS
Ques: Simplify the rational expression (x2 + 7x + 10) / (x2 – 4). (3 marks)
Ans: Factor both the top and bottom of the expression.
= (x2 + 7x + 10) / (x2 – 4) ? (x + 5) (x + 2) / (x2 – 22)
? (x + 5) (x + 2) / (x + 2) (x – 2)
Cancel the common terms to get;
= (x + 5) / (x – 2)
Ques: Simplify: (6x2 – 54) / (x2 + 7x + 12). (3 marks)
Ans: (6x2 – 54) / (x2 + 7x + 12)
= 6(x2 – 9) / (x + 3) (x + 4)
= 6(x2 – 32) / (x + 3) (x + 4)
= 6(x + 3) (x – 3) / (x + 3) (x + 4)
= 6(x – 3)/(x + 4)
Ques: Prove (9p – 5q)2 + 180pq = (9p + 5q)2 . (4 marks)
Ans: LHS = (9p – 5q)2 + 180 pq
= {(9p)2 – 2(9p) (5q) + (5q)2} + 180 pq
= 81p2 – 90pq + 25p2 + 180 pq
= 81p2 + 90pq + 25q2
= (9p)2 + 2(9p)(5q) + (5q)2
= (9p + 5q)2 = RHS.
Ques: Subtract 3a (a + b + c) - 2b (a - b + c) from 4c (-a + b + c). (4 marks)
Ans: 4c (-a + b + c) - {3a (a + b + c) - 2b (a - b + c)}
= {4c × (-a) + 4c × b + 4c × c} - {(3a × a + 3a × b + 3a × c)} + {(-2b × a) + (-2b × -b) + (-2b × c)}
= {-4ac + 4bc + 4c2} - {3a2 + 3ab + 3ac - 2ab + 2b2 - 2bc}
= -4ac + 4bc + 4c2 - 3a2 - 3ab - 3ac + 2ab - 2b2 + 2bc
= -3a2 - 2b2 + 4c2 - 7ac + 6bc - ab
Ques: Find the length of a rectangular tabletop whose area is given as (x2 + 4x − 5) sq. units and breadth are (x − 1) units. (4 marks)
Ans: Length of a Rectangle = Area / Breadth
Length = (x2 + 4x - 5) / (x - 1)
Length = (x + 5) (x - 1) / (x - 1)
Length = (x + 5) units
Length of the top is (x + 5) units
Ques: Simplify (x−1) (x2−2x−3) / (x−3) (x2−5x+4). (4 marks)
Ans: Let, f(x) = (x−3) (x2−5x+4)
= (x−3) (x−1) (x−4)
And g(x) = (x−1) (x2−2x−3)
= (x−1) (x−3) (x+1)
Thus, g(x)/ f(x)= (x-1) (x-3) (x+1) / (x-3) (x-1) (x-4) = (x+1) / (x−4)
Ques: Simplify (xy + 3z)2. (2 marks)
Ans: (xy + 2z)2
= (xy)2 + 2(xy)(2z) + (2z)2 {Using: (a + b)2 = a2 + 2ab + b2}
= x2y2 + 4xyz + 4z2
Ques: Show that (3x + 5)2 - 60x = (3x - 5)2 .(4 marks)
Ans: LHS = (3x + 5)2 - 60x
= (9x2 + 30x + 25) - 60x
= 9x2 - 30x + 25
= (3x)2 - 2(3x)(5) + (5)2
= (3x - 5)2 = RHS
Ques: Subtract 3a (a + b) - 2b (a - b) from 4c (-a + b). (4 marks)
Ans: 4c (-a + b) - {3a (a + b) - 2b (a - b)}
= {4c × (-a) + 4c × b} - {(3a × a + 3a × b} + {(-2b × a) + (-2b × -b)}
= {-4ac + 4bc} - {3a2 + 3ab - 2ab + 2b2}
= -4ac + 4bc - 3a2 - 3ab + 2ab - 2b2
= -3a2 - 2b2 - 4ac + 4bc - ab
Ques: Find the length of a rectangular tabletop whose area is given as (x2 + 2x − 1) sq. units and breadth are (x − 1) units. (4 marks)
Ans: Length of a Rectangle = Area / Breadth
Length = (x2 + 2x - 1) / (x - 1)
Length = (x + 1) (x - 1) / (x - 1)
Length = (x + 1) units
Length of the top is (x + 1) units
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