A rational expression is simply an algebraic fraction with polynomials as its numerator and denominator. In this article, we will explore the concepts of adding and subtracting these expressions.
Just like regular numerical fractions, the key to adding and subtracting rational expressions is having the same denominators. This rule applies to rational expressions as well.
When dealing with expressions that have like denominators, simply add or subtract (depending on the sign) the numerators while keeping the denominator constant.
Note: When a polynomial is multiplied by a negative sign, all of its terms need to be negated. For example, -(2x-1) is equivalent to -2x+1.
Example: If we have the common denominator (x-1), we can simplify (x+1) + (2x-1) to get 3x.
If the denominators are different, we must first manipulate them to make them the same before adding or subtracting. To do this, follow these steps:
Example: For the denominators 4x²-y² and 2x+y, we can simplify the expression by multiplying the first denominator by (2x-y) to get (2x+y)(2x-y). The resulting LCM is 4x²-y², and the expression simplifies to 3x-1.
When dealing with polynomial denominators, we need to find the LCM (Least Common Multiple) to add or subtract them. The process is similar to finding the LCM of integers. Here's an example:
Find the LCM of 15mn and 21np².
Solution: First, break each term into its prime factors and smallest variable factors. Then, multiply each factor by the greatest number of times it appears in any of the factorizations to get the LCM, which in this case is 105mnp².
Key Takeaways:
How can rational algebraic expressions be solved through addition and subtraction?
To solve expressions with like denominators, simply add or subtract the numerators while keeping the denominator constant. For expressions with different denominators, first find the LCM of the denominators and then add or subtract the new numerators over the common denominator.
What is the addition and subtraction of rational expressions?
The addition and subtraction of rational expressions involve combining two or more algebraic fractions that have polynomials as numerator and denominator. The final expression is then simplified, if possible.
Rational expressions, or fractions with variables in the numerator or denominator, can be added or subtracted in a similar way to regular numerical fractions. However, there are specific rules and methods to follow to ensure the correct solution is obtained. In this article, we will take an in-depth look at these rules and methods for adding and subtracting rational expressions with different denominators.
Example: If we have the denominators 2x+3 and 3x+4, we can find the LCM by multiplying the first denominator by (3x+4) to get (2x+3)(3x+4). The resulting LCM is (2x+3)(3x+4), and the expression simplifies to 5x+7.
Key Takeaways:
How can rational expressions be solved through addition and subtraction?
To solve expressions with like denominators, simply add or subtract the numerators while keeping the denominator constant. For expressions with different denominators, first find the LCM of the denominators and then add or subtract the new numerators over the common denominator.
What is the addition and subtraction of rational expressions?
The addition and subtraction of rational expressions involve combining two or more fractions with variables in the numerator or denominator, and then simplifying the final expression, if possible.
Adding and subtracting rational expressions can seem daunting at first, but with the right approach, it can become much simpler. The key is to first ensure that the denominators are equal, which allows for simpler addition or subtraction of the expressions. It's worth noting that factoring and simplifying the denominators before finding the LCM can also make the process easier.
Now that you understand the rules and methods for adding and subtracting rational expressions, it's important to keep practicing. With practice, you'll soon build confidence in your ability to solve these types of equations effortlessly.
