Problem 43 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{2 x}{x+2}-\frac{8}{x^{2}+2 x}+\frac{3}{x}$$

Step-by-Step Solution

Answer

\(\frac{2x^2 + 3x - 2}{x(x+2)}\)

Step 1: Identify a Common Denominator

The denominators in the expression are \(x+2\), \(x^2+2x\), and \(x\). We need a common denominator to simplify the expression. Notice that \(x^2+2x = x(x+2)\). Therefore, the common denominator is \(x(x+2)\).

Step 2: Rewrite Each Fraction with Common Denominator

Rewrite each term with the common denominator \(x(x+2)\):- The first term becomes \(\frac{2x}{x+2} = \frac{2x \cdot x}{x(x+2)} = \frac{2x^2}{x(x+2)}\).- The second term is already with the common denominator: \(\frac{8}{x^2 + 2x} = \frac{8}{x(x+2)}\).- The third term becomes \(\frac{3}{x} = \frac{3(x+2)}{x(x+2)} = \frac{3x + 6}{x(x+2)}\).

Step 3: Combine the Fractions

Combine the fractions with the common denominator:\[ \frac{2x^2}{x(x+2)} - \frac{8}{x(x+2)} + \frac{3x + 6}{x(x+2)} \]

Step 4: Simplify the Numerator

Combine the numerators:\[ 2x^2 - 8 + 3x + 6 = 2x^2 + 3x - 2 \]

Step 5: State the Simplified Expression

The simplified fraction is:\[ \frac{2x^2 + 3x - 2}{x(x+2)} \]

Key Concepts

Common Denominator

When working with rational expressions, finding a common denominator is crucial. This step allows us to add or subtract fractions by ensuring they have the same denominator. It often involves identifying the least common multiple (LCM) of the denominators given in the expressions.

• Start by looking at each denominator separately.
• If the denominators involve polynomials, factor them where possible.
• Find the LCM by selecting the highest power of each factor present.

In the exercise example, we dealt with the denominators \(x+2\), \(x^2 + 2x\), and \(x\). Noticing that \(x^2 + 2x = x(x+2)\) reveals that the common denominator is \(x(x+2)\). This common ground allows the fractions to be rewritten, paving the way for combining them.

Simplifying Algebraic Fractions

Simplifying algebraic fractions may seem challenging but can be straightforward with a consistent approach. The goal is to make the expression as simple as possible by reducing common factors in the numerator and denominator.

• Begin with rewriting terms with a common denominator.
• Combine the numerators into a single expression.
• Simplify the resulting expression by factoring if possible.

In our exercise, we expressed each term with a common denominator of \(x(x+2)\), combining them to form a single fraction \(\frac{2x^2 + 3x - 2}{x(x+2)}\). Check if the numerator can be factored further, potentially simplifying the expression even more. This might not always be possible, but it's worth looking into.

Polynomials

Understanding polynomials is fundamental for dealing with rational expressions as they often form both numerators and denominators. A polynomial is an expression consisting of variables and coefficients, involving operations of addition, subtraction, and multiplication.

• Identify the degree of the polynomial, which is the highest power of the variable involved.
• Factor polynomial expressions when possible, as this can help in simplifying expressions or finding common denominators.
• Use polynomial identities and formulas to assist in manipulation.

In the exercise, recognizing that \(x^2 + 2x\) factors into \(x(x+2)\) helped set the stage for finding a common denominator. This highlights how factoring polynomials is a key skill in simplifying rational expressions.