Problem 5
Question
Add or Subtract the following rational expressions. $$ \frac{4 x^{2}-x+1}{3 x+10}-\frac{x^{2}+2 x+5}{3 x+10} $$
Step-by-Step Solution
Verified Answer
Question: What is the simplified form of the given expression:
$$
\frac{4x^2 - x + 1}{3x + 10} - \frac{x^2 + 2x + 5}{3x + 10}
$$
Answer: The simplified form of the given expression is:
$$
\frac{3x^2 - 3x - 4}{3x + 10}
$$
1Step 1: Identify the denominators and numerators
The given rational expression is:
$$
\frac{4x^2 - x + 1}{3x + 10} - \frac{x^2 + 2x + 5}{3x + 10}
$$
Here, the numerator of the first rational expression is \(4x^2 - x + 1\), and the denominator is \(3x + 10\). The numerator of the second rational expression is \(x^2 + 2x + 5\), and the denominator is \(3x + 10\).
2Step 2: Add or subtract the numerators
Since the denominators are the same, we can add (or subtract) the numerators directly. In this case, we need to subtract the numerators of the two rational expressions.
$$
(4x^2 - x + 1) - (x^2 + 2x + 5)
$$
3Step 3: Simplify the expression
Now, we have to simplify the expression obtained in Step 2.
$$
4x^2 - x + 1 - x^2 - 2x - 5
$$
Combine the like terms:
$$
(4x^2 - x^2) + (-x - 2x) + (1 - 5)
$$
$$
3x^2 - 3x - 4
$$
4Step 4: Write the final expression
Now that we have the simplified numerator, we can write the final rational expression by placing the simplified numerator over the common denominator.
$$
\frac{3x^2 - 3x - 4}{3x + 10}
$$
The final expression is:
$$
\frac{3x^2 - 3x - 4}{3x + 10}
$$
Key Concepts
Rational ExpressionsCommon DenominatorSimplifying Algebraic ExpressionsCombining Like Terms
Rational Expressions
Rational expressions are fractions that have polynomials in both the numerator and the denominator. Just like when dealing with regular fractions, the key operations involving rational expressions include addition, subtraction, multiplication, and division.
When adding or subtracting rational expressions, it is crucial to find a common denominator to combine them effectively. If the denominators are already the same, as in the example exercise, the process becomes much simpler, as you can combine the numerators directly.
When adding or subtracting rational expressions, it is crucial to find a common denominator to combine them effectively. If the denominators are already the same, as in the example exercise, the process becomes much simpler, as you can combine the numerators directly.
Common Denominator
When working with addition or subtraction of fractions, having a common denominator is essential. This concept also applies to rational expressions. The common denominator allows us to perform addition or subtraction on the numerators while keeping the denominator consistent.
In our example, since the denominators are both \(3x + 10\), we did not need to manipulate them to find a common denominator. However, if they had been different, we would have needed to find a least common multiple of the denominators first, creating equivalent rational expressions with a shared denominator before we could proceed with adding or subtracting.
In our example, since the denominators are both \(3x + 10\), we did not need to manipulate them to find a common denominator. However, if they had been different, we would have needed to find a least common multiple of the denominators first, creating equivalent rational expressions with a shared denominator before we could proceed with adding or subtracting.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is all about combining like terms and reducing the expression to its simplest form. Like terms are terms that have the same variables raised to the same power. For instance, \(4x^2\) and \(x^2\) are like terms. Simplification may include adding or subtracting like terms as well as multiplying or dividing by constant factors.
In the provided exercise, simplification involved subtracting \(x^2\) from \(4x^2\), and combining \( - x - 2x \) and constant terms \(1 - 5\), which streamlined the expression down to \(3x^2 - 3x - 4\).
In the provided exercise, simplification involved subtracting \(x^2\) from \(4x^2\), and combining \( - x - 2x \) and constant terms \(1 - 5\), which streamlined the expression down to \(3x^2 - 3x - 4\).
Combining Like Terms
Combining like terms is an essential step in algebra that helps in simplifying expressions. Like terms have the same variable components and exponents. To combine them, simply add or subtract their coefficients.
For instance, in our problem, the like terms were combined as follows: \((4x^2 - x^2)\), \((- x - 2x)\), and \((1 - 5)\). Each of these operations involves combining terms with the same variable to the same power (or no variable in the case of constant terms), resulting in the simplified expression \(3x^2 - 3x - 4\).
For instance, in our problem, the like terms were combined as follows: \((4x^2 - x^2)\), \((- x - 2x)\), and \((1 - 5)\). Each of these operations involves combining terms with the same variable to the same power (or no variable in the case of constant terms), resulting in the simplified expression \(3x^2 - 3x - 4\).
Other exercises in this chapter
Problem 5
Find the domain of each of the following rational expressions. $$ \frac{12 y}{3 y^{2}-2 y-8} $$
View solution Problem 5
Perform each multiplication. $$ \frac{x^{2}+6 x+8}{x^{2}-6 x+8} \cdot \frac{x^{2}-2 x-8}{x^{2}+2 x-8} $$
View solution Problem 5
Reduce each of the following fractions to lowest terms. $$ \frac{-3 a^{4}+75 a^{2}}{2 a^{3}-16 a^{2}+30 a} $$
View solution Problem 6
For the following problems, find the domain of each rational expression. $$ \frac{5 r+6}{9 r(2 r+1)} $$
View solution