Problem 8 - College Algebra with Modeling and Visualization

Question

Divide the expression. $$\frac{\left(1-5 x^{2}\right)(x+1)+x^{2}}{2 x}$$

Step-by-Step Solution

Answer

Simplified expression: \(- \frac{5x^2}{2} - 2x + \frac{1}{2} + \frac{1}{2x}\)."

Step 1: Identify and Simplify the Numerator

The given expression is \( \frac{(1-5x^2)(x+1) + x^2}{2x} \). Start by simplifying the numerator. Distribute \((1-5x^2)\) across \((x+1)\), which gives you: \((1)(x) + (1)(1) - (5x^2)(x) - (5x^2)(1)\). Compute each multiplication to get \(x + 1 - 5x^3 - 5x^2\). Now add the \(x^2\) term from the original equation to yield \(x + 1 - 5x^3 - 5x^2 + x^2\).

Step 2: Combine Like Terms

Combine the like terms in the simplified numerator from Step 1. Combine \(-5x^2 + x^2\) to get \(-4x^2\). Now rewrite the expression as \(x + 1 - 5x^3 - 4x^2\).

Step 3: Simplify the Fraction

Now, divide each term in the polynomial by the denominator \(2x\). This results in splitting the fraction: \(\frac{x}{2x} + \frac{1}{2x} - \frac{5x^3}{2x} - \frac{4x^2}{2x}\). Simplify each fraction: \(\frac{x}{2x} = \frac{1}{2}\), \(\frac{1}{2x}\) stays as is, \(\frac{5x^3}{2x} = \frac{5x^2}{2}\), and \(\frac{4x^2}{2x} = 2x\). This gives \(\frac{1}{2} + \frac{1}{2x} - \frac{5x^2}{2} - 2x\).

Step 4: Write Final Expression

Combine and organize the terms obtained from simplifying the fraction. The final simplified expression is: \(- \frac{5x^2}{2} - 2x + \frac{1}{2} + \frac{1}{2x}\).

Key Concepts

Numerator Simplification

Simplifying the numerator is crucial when dealing with polynomial division. To simplify the numerator of our original expression \( \frac{(1-5x^2)(x+1) + x^2}{2x} \), start with distribution. The expression inside the parentheses, \((1-5x^2)(x+1)\), requires expanding each term.
1. Multiply \(1\) by each component of \((x+1)\), resulting in \(1 \cdot x + 1 \cdot 1\).2. Similarly, multiply \(-5x^2\) by each part of \((x+1)\), giving \(-5x^2 \cdot x - 5x^2 \cdot 1\).
This yields the expanded form: \(x + 1 - 5x^3 - 5x^2\). Remember to incorporate \(+ x^2\) at the end as per the original expression.
As a result, the complete simplified numerator is \(x + 1 - 5x^3 - 5x^2 + x^2\).

Combining Like Terms

After simplifying the numerator, the next step is to combine like terms. When you have a polynomial, like terms are the terms that have the same variable raised to the same power.In our expression \(x + 1 - 5x^3 - 5x^2 + x^2\), the like terms \(-5x^2\) and \(+ x^2\) can be combined. Simply add the coefficients:

• \(-5x^2 + x^2 = -4x^2\)

Thus, after combining like terms, the numerator becomes \(x + 1 - 5x^3 - 4x^2\). This simplification makes further calculations easier by reducing complexity.

Fraction Simplification

Once the numerator is simplified and the like terms combined, the expression should be divided by the denominator. In this scenario, divide each term in \(x + 1 - 5x^3 - 4x^2\) by \(2x\).
Break down the fraction into separate terms:

• \(\frac{x}{2x} = \frac{1}{2}\)
• \(\frac{1}{2x}\) remains unchanged
• \(\frac{-5x^3}{2x} = \frac{-5x^2}{2}\)
• \(\frac{-4x^2}{2x} = -2x\)

Each term of the fraction is simplified individually, creating an expression that is much easier to manage and understand.

Step by Step Algebra

Breaking down complex algebraic expressions into smaller, more manageable parts is essential for solving them accurately. The process in this exercise is a great demonstration of step-by-step algebra, where each move builds upon the last.1. **Numerator Simplification** - Distributing and expanding requires careful attention but simplifies the expression significantly.2. **Combining Like Terms** - Reducing terms by summing their coefficients aids in clarity and simplifies subsequent operations.3. **Fraction Simplification** - Dividing each term by the denominator makes it possible to handle complex fractions easily.
By addressing each part sequentially, you can simplify even the most daunting algebraic expressions methodically and clearly. This approach ensures each step is understandable, leading to a final refined expression: \(- \frac{5x^2}{2} - 2x + \frac{1}{2} + \frac{1}{2x}\).