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Chapter 6: Problem 35

Write each rational expression in lowest terms. $$ \frac{3 x+7}{3} $$

Short Answer

Expert verified
The rational expression \( \frac{3x+7}{3} \) is already in its lowest terms.

Step by step solution

01

Identify the rational expression

The given rational expression is \( \frac{3x+7}{3} \).
02

Factorize the numerator and denominator

Both the numerator \(3x + 7\) and the denominator \(3\) are already in their simplest forms. There are no common factors between the terms in the numerator and the denominator that can be cancelled out.
03

Simplify (if possible)

Since there are no common factors other than 1, the expression \( \frac{3x+7}{3} \) is already in its lowest terms. Therefore, it cannot be simplified any further.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Polynomials
Polynomials are mathematical expressions that involve the sum of powers of variables. Factoring polynomials is a crucial step when simplifying rational expressions. It means rewriting the polynomial as a product of its factors, which can help reveal common terms that may cancel out.
For instance, consider the polynomial in the following rational expression: \(\frac{3x+7}{3}\) .
The numerator, \(3x + 7\), is already in its simplest form because it cannot be factored any further using integers.
The denominator is a constant, \(3\), which remains as is. Since there are no common factors between the numerator and the denominator, they cannot be simplified further.
This step helps ensure that we reduce the expression to its lowest terms later on.
Lowest Terms
Simplifying a rational expression to its lowest terms means reducing it to its simplest form. This is done by dividing both the numerator and the denominator by their common factors.
In the given expression \(\frac{3x+7}{3}\), we check for common factors between the numerator \(3x + 7\) and the denominator \(3\).

Since there are no common factors other than 1, the expression is already in its lowest terms. This makes it easier to understand and work with.
Here are the usual steps to ensure a rational expression is in the lowest terms:
  • Factor both the numerator and the denominator completely.
  • Identify and cancel out common factors.
  • Rewrite the simplified expression.

Applying these steps ensures the rational expression is as simple as possible.
Rational Expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. Rational expressions follow the same rules as regular fractions, including simplification and operations like addition and multiplication.
The given rational expression is \(\frac{3x+7}{3}\). Here, the numerator is a polynomial \(3x + 7\) and the denominator is a constant \(3\).

The goal is to simplify it by factoring and reducing to lowest terms. Rational expressions are simplified by factoring out and cancelling common terms just like regular fractions:
  • When both parts of the expression share a factor, it can be cancelled out.
Understanding these concepts can help with more complex algebraic manipulations and solving equations involving rational expressions.

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Most popular questions from this chapter

Write each rational expression in lowest terms. $$ \frac{p^{2}+q^{2}}{p^{2}-q^{2}} $$

Multiply or divide as indicated. $$ \frac{(-3 m n)^{2} \cdot 64\left(m^{2} n\right)^{3}}{16 m^{2} n^{4}\left(m n^{2}\right)^{3}} \div \frac{24\left(m^{2} n^{2}\right)^{4}}{\left(3 m^{2} n^{3}\right)^{2}} $$

Multiply or divide as indicated. $$ \frac{12 x-20}{5 x} \cdot \frac{6}{9 x-15} $$

Write each rational expression in lowest terms. $$ \frac{x^{2}(x+1)}{x(x+1)} $$

Which two rational expressions are equivalent to \(-1 ?\) A. \(\frac{2 x+3}{2 x-3}\) B. \(\frac{2 x-3}{3-2 x}\) C. \(\frac{2 x+3}{3+2 x}\) D. \(\frac{2 x+3}{-2 x-3}\)
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