Embark on an intellectual journey with our comprehensive guide to adding and subtracting rational expressions quick check. Delve into the intricacies of rational expressions, mastering the art of simplifying, combining, and applying them in real-world scenarios. Prepare to elevate your mathematical prowess with this authoritative exploration.
As we delve into the topic, we will unravel the fundamental concepts of rational expressions, empowering you with the tools to simplify and manipulate them with confidence. Through step-by-step procedures and illustrative examples, you will gain a thorough understanding of the rules governing the addition and subtraction of rational expressions, both with like and unlike denominators.
Understanding Rational Expressions
Rational expressions are algebraic expressions that represent the quotient of two polynomials. They are made up of a numerator and a denominator, which are both polynomials. For example, the rational expression 3x / (x – 2) has a numerator of 3x and a denominator of x – 2.
Rational expressions can have different degrees, depending on the degree of the numerator and the degree of the denominator. The degree of a rational expression is the highest exponent of the variable in the expression. For example, the rational expression 3x / (x – 2) has a degree of 1, because the highest exponent of the variable x is 1.
Simplifying Rational Expressions
Simplifying rational expressions involves reducing them to their simplest form. This can be done by factoring the numerator and denominator, canceling common factors, and reducing fractions to their simplest form.
For example, to simplify the rational expression 3x / (x – 2), we can first factor the numerator and denominator:
3x = 3 – x
(x – 2) = (x – 2)
Then, we can cancel the common factor of x:
3x / (x – 2) = (3 – x) / (x – 2) = 3 / 1 = 3
Therefore, the simplified form of the rational expression 3x / (x – 2) is 3.
Adding and Subtracting Rational Expressions
To add or subtract rational expressions, we need to make sure that they have the same denominator. This can be done by finding the least common multiple (LCM) of the denominators.
For example, to add the rational expressions 1 / x + 1 / y, we first need to find the LCM of x and y. The LCM of x and y is xy. Therefore, we can rewrite the rational expressions as follows:
1 / x = x / xy
1 / y = y / xy
Now, we can add the rational expressions:
1 / x + 1 / y = x / xy + y / xy = (x + y) / xy
Therefore, the sum of the rational expressions 1 / x + 1 / y is (x + y) / xy.
Applications of Rational Expressions
Rational expressions have many applications in real life. For example, they are used in physics to calculate the speed of a moving object, in engineering to design bridges and buildings, and in economics to model the supply and demand of goods and services.
Here are some examples of problems that involve adding or subtracting rational expressions:
- A train travels 100 miles in 2 hours. What is the average speed of the train?
- A company produces 100 widgets per hour. How many widgets will the company produce in 5 hours?
- A store sells apples for $1 each and oranges for $0.50 each. How much will it cost to buy 3 apples and 2 oranges?
Advanced Topics, Adding and subtracting rational expressions quick check
In addition to the basic operations of addition and subtraction, rational expressions can also be multiplied and divided. Multiplying rational expressions is straightforward, but dividing rational expressions requires the use of the reciprocal rule.
For example, to multiply the rational expressions 1 / x and 1 / y, we simply multiply the numerators and the denominators:
1 / x – 1 / y = (1 – 1) / (x – y) = 1 / xy
To divide the rational expressions 1 / x and 1 / y, we use the reciprocal rule:
1 / x / 1 / y = 1 / x – y / 1 = y / x
Detailed FAQs: Adding And Subtracting Rational Expressions Quick Check
What are the steps involved in adding rational expressions with like denominators?
1. Add the numerators while keeping the denominator the same.
2. Simplify the resulting expression if possible.
How do I subtract rational expressions with unlike denominators?
1. Find the least common multiple (LCM) of the denominators.
2. Rewrite each expression with the LCM as the denominator.
3. Subtract the numerators while keeping the denominator the same.
4. Simplify the resulting expression if possible.
