Problem 124 - Elementary and Intermediate Algebra

Question

Perform each operation. $$ \left(2 x^{2}-9 x-5\right) \cdot \frac{x}{2 x^{2}+x} $$

Step-by-Step Solution

Answer

\( \frac{2x^2 - 9x - 5}{2x + 1} \) is the simplified result of the operation.

Step 1: Rewrite the Expression

The given expression is \( \left(2x^2 - 9x - 5\right) \cdot \frac{x}{2x^2 + x} \). First, notice that the second part is a fraction. Rewrite the expression as a multiplication of fractions: \[ \frac{2x^2 - 9x - 5}{1} \cdot \frac{x}{2x^2 + x} \].

Step 2: Factor the Denominator

The denominator of the second fraction is \(2x^2 + x\). Factor out the greatest common factor, which is \(x\), to get \(x(2x + 1)\). So the expression is now \[ \frac{2x^2 - 9x - 5}{1} \cdot \frac{x}{x(2x + 1)} \].

Step 3: Simplify the Rational Expression

With the expression now as \( \frac{2x^2 - 9x - 5}{1} \cdot \frac{x}{x(2x + 1)} \), cancel the common factor of \(x\) from the numerator of the second fraction and the denominator, yielding \( \frac{2x^2 - 9x - 5}{1} \cdot \frac{1}{2x + 1} \).

Step 4: Perform the Multiplication

Multiply the simplified numerators and denominators: \[ \frac{(2x^2 - 9x - 5) \cdot 1}{1 \cdot (2x + 1)} = \frac{2x^2 - 9x - 5}{2x + 1} \]. This is the final expression.

Key Concepts

Factoring

Factoring is an essential technique in algebra that simplifies expressions, making them easier to work with. It involves breaking down a complex expression into simpler components, called factors, which when multiplied together give back the original expression. In the given exercise, we primarily focused on factoring the denominator of the rational expression.

To factor an expression, identify the greatest common factor (GCF) that can be extracted from each term. For instance, in the expression \(2x^2 + x\), the terms share a common factor of \(x\). By factoring out \(x\), the expression becomes \(x(2x + 1)\). This step is crucial to simplifying the subsequent rational expression.

• Always look for a GCF first before proceeding with complex factoring.
• Remember, factoring might involve recognizing patterns like difference of squares or trinomials.
• Factoring simplifies expressions and reveals possible cancellations in rational expressions.

Rational expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying these expressions involves both factoring and canceling out common terms to reduce the expression to its simplest form.

For our exercise, the rational expression involved is \( \frac{x}{2x^2 + x} \). As described in the factoring section, the denominator was factored to \(x(2x + 1)\). This allows any common factors between the numerator and denominator to be canceled, simplifying the expression to \(\frac{1}{2x+1}\) when the common factor \(x\) is removed.

• Make sure to factor both the numerator and the denominator completely.
• Cancel any common factors to simplify the rational expression.
• The goal is to achieve the simplest possible form to ease further operations.

This process makes it more intuitive and reduces the complexity of further multiplying the rational expressions.

Polynomial multiplication

Polynomial multiplication is the process of distributing each term of one polynomial to every term of another. This results in a new polynomial that represents the product. The challenge lies in ensuring that each term is correctly multiplied and combined.

In the simplified step of our exercise, the expression \((2x^2 - 9x - 5) \cdot 1\) was straightforward due to the multiplication with 1, but the setup stresses the importance of understanding this process.

• Distribute each term in the first polynomial to each term in the second.
• Combine like terms after distribution for simplification.
• Remember, multiplying polynomials increases the degree based on the terms involved.

In this exercise, the multiplication step was simple because the expression was already in its basic form. Yet understanding this process is crucial for handling more complicated polynomial products in future problems.