Problem 45 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$3+\frac{5}{u}+\frac{2 u}{3 u+1}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{2u^2 + 24u + 8}{u(3u+1)} \).

Step 1: Simplify the Middle Fraction

The term \( \frac{5}{u} \) is already simplified as much as possible assuming \( u eq 0 \). There is nothing more to do with this fraction as it is in its simplest form.

Step 2: Simplify the Last Fraction

The term \( \frac{2u}{3u+1} \) involves a polynomial in both the numerator and the denominator. It is already in its simplest form since we cannot factor further unless there is a common factor or further instructions are given.

Step 3: Write the Expression with a Common Denominator

To simplify the overall expression, a common denominator is needed for the fractions. We use the least common multiple of \( u \) and \( 3u+1 \), which is \( u(3u+1) \). We rewrite each term with this common denominator.

Step 4: Rewrite Each Term

1. Rewrite \( 3 \) as \( \frac{3(3u+1)}{u(3u+1)} = \frac{9u+3}{u(3u+1)} \).2. Rewrite \( \frac{5}{u} \) as \( \frac{5(3u+1)}{u(3u+1)} = \frac{15u+5}{u(3u+1)} \).3. Rewrite \( \frac{2u}{3u+1} \) as \( \frac{2u}{3u+1} \cdot \frac{u}{u} = \frac{2u^2}{u(3u+1)} \).

Step 5: Combine All Fractions

Now, sum the numerators over the common denominator:\[ \frac{9u+3}{u(3u+1)} + \frac{15u+5}{u(3u+1)} + \frac{2u^2}{u(3u+1)} = \frac{2u^2 + 9u + 15u + 3 + 5}{u(3u+1)} \]

Step 6: Simplify the Numerator

Combine like terms in the numerator:\[ 2u^2 + 9u + 15u + 3 + 5 = 2u^2 + 24u + 8 \]Thus, the expression becomes: \[ \frac{2u^2 + 24u + 8}{u(3u+1)} \].

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, structured together using operations of addition, subtraction, multiplication, and non-negative integer exponents. Understanding polynomials is crucial in solving many algebraic problems. In the expression \(3 + \frac{5}{u} + \frac{2u}{3u+1}\), the term \(3u+1\) in the denominator of the last fraction is a simple polynomial because it involves the variable \(u\) raised to the first power and constant terms. The structure of polynomials can vary:

• Monomial: A single term such as \(5u\) or \(3\).
• Binomial: Two terms like \(3u + 1\).
• Trinomial: Three terms, though \(3u + 1\) here is not an example, it's just to illustrate a mix of three polynomial terms.

Polynomials can be simplified by combining like terms, factoring, or applying distributive properties when needed. In this example, simplifying the polynomial in the denominators allows for easier manipulation and helps in finding a common denominator for all terms.

Common Denominator

Finding a common denominator is essential when adding or subtracting rational expressions with different denominators. The common denominator represents a shared base for the fractions, making it possible to merge them into a single fraction. In our expression \(3 + \frac{5}{u} + \frac{2u}{3u+1}\), the task is to find a common base to rewrite each term in the expression.To determine a common denominator:

• Identify the denominators in the expression: here, they are \(u\) and \(3u + 1\).
• Find the least common multiple (LCM) of these denominators, which in this instance is \(u(3u+1)\).
• Rewrite each fractional term with this new common denominator to prepare for further simplification.

Once rewritten, the expressions like \(\frac{5}{u}\) become \(\frac{15u + 5}{u(3u+1)}\), translating \(\frac{2u}{3u+1}\) into \(\frac{2u^2}{u(3u+1)}\), thus aligning all terms under a single platform. Simplifying fractions to a common denominator makes it possible to algebraically operate over the entire expression, such as performing sums or differences.

Rational Expressions

Rational expressions are fractions in which the numerator and the denominator are polynomials. Simplifying rational expressions, like \(\frac{2u^2 + 24u + 8}{u(3u+1)}\), involves several steps, much like simplifying numerical fractions. The process includes:

• Reducing to the simplest form by factoring polynomials, if possible.
• Finding and applying a common denominator to enable addition or subtraction of the fractions involved.
• Combining like terms by adding the numerators after rewriting each fraction accordingly.

In our exercise, once we find the common denominator \(u(3u+1)\), we can add the numerators of all terms \(9u + 3 + 15u + 5 + 2u^2\), combining two linear terms \(9u\) and \(15u\) into \(24u\), alongside constants. This results in a simplified rational expression. Rational expressions simplify complex algebraic expressions, making it easier to handle equations in mathematics.