Problem 27 - Algebra for College Students

Question

For Problems 27-30, solve each equation by first applying the multiplication property of equality to produce an equivalent equation with integral coefficients. $$ \frac{1}{10} x^{3}+\frac{1}{5} x^{2}-\frac{1}{2} x-\frac{3}{5}=0 $$

Step-by-Step Solution

Answer

The equivalent equation is \( x^3 + 2x^2 - 5x - 6 = 0 \).

Step 1: Identify the Least Common Denominator

Identify the least common denominator (LCD) for all the fractions in the equation. In this case, the denominators are 10, 5, and 2. The LCD for these numbers is 10.

Step 2: Multiply Each Term by the LCD

Multiply each term of the equation by the least common denominator (10) to eliminate the fractions:\[ 10 \times \left( \frac{1}{10} x^3 \right) + 10 \times \left( \frac{1}{5} x^2 \right) - 10 \times \left( \frac{1}{2} x \right) - 10 \times \left( \frac{3}{5} \right) = 0 \] Simplifying, we get:\[ x^3 + 2x^2 - 5x - 6 = 0 \]

Step 3: Verify the New Equation

Check all the terms of the new equation to ensure that all coefficients are integers. The new equation is \( x^3 + 2x^2 - 5x - 6 = 0 \), and all coefficients are integers.

Step 4: Solve the Integral Equation

While the problem asks only for transforming the equation, to solve, use methods like factoring, synthetic division, or numerical techniques to find the roots of \( x^3 + 2x^2 - 5x - 6 = 0 \).

Key Concepts

Least Common Denominator

When working with equations that include fractions, it is often necessary to find a common ground to simplify them. This is where the Least Common Denominator (LCD) comes into play. The LCD is the smallest number that all the given denominators can divide into without a remainder.

• Identify all the denominators in the equation.
• Determine the smallest number each denominator can evenly divide into.

In our example, we have the denominators 10, 5, and 2. The LCD for these numbers is 10.
By identifying the LCD, we set the stage to eliminate the fractions and simplify the equation, making subsequent calculations much simpler.

Integral Coefficients

Once the least common denominator is identified, we aim to transform the equation so that all coefficients become integers. This step is crucial since working with whole numbers is much simpler and less error-prone than dealing with fractions.
When multiplying each term by the LCD, we remove the fractions, transforming their denominators into 1. For instance, multiplying \( \frac{1}{10} x^3 \) by 10 results in \( x^3 \), an integral coefficient. This transformation leads to an equation like \[ x^3 + 2x^2 - 5x - 6 = 0 \]where each term is composed of integral coefficients.

Polynomial Equation

A polynomial equation is simply an equation that involves only the addition, subtraction, multiplication, and non-negative integer exponents of variables. The simplified form of our initial equation, \( x^3 + 2x^2 - 5x - 6 = 0 \), is an example of a cubic polynomial equation.
Key characteristics of polynomial equations:

• The highest exponent is known as the degree of the polynomial.
• Polynomial without fractions is often easier to solve or manipulate mathematically.

Understanding polynomial equations is critical because they form the foundation of algebra and many higher mathematical concepts.

Fraction Elimination

Fraction elimination is a technique often used to simplify equations and is particularly helpful when these equations involve fractions. By multiplying through by the least common denominator, we effectively remove all fractional components, which simplifies both solving and understanding the equation.
Here's how it works:

• Multiply each term in the equation by the LCD.
• Simplify each term to ensure that no fractions remain in the equation.

Once done, the equation becomes another form with integral coefficients, ready for further solving techniques such as factoring or using the quadratic formula if applicable.