Problem 28 - 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. $$ \frac{1}{10} x^{3}+\frac{1}{2} x^{2}+\frac{1}{5} x-\frac{4}{5}=0 $$

Step-by-Step Solution

Answer

The equivalent equation with integral coefficients is \( x^3 + 5x^2 + 2x - 8 = 0 \).

Step 1: Identify Fractional Coefficients

First, identify all the fractions in the given equation: \( \frac{1}{10} x^{3} \), \( \frac{1}{2} x^{2} \), \( \frac{1}{5} x \), and \( -\frac{4}{5} \). The denominators are 10, 2, and 5.

Step 2: Determine the Least Common Denominator (LCD)

Find the least common multiple of the denominators 10, 2, and 5. The smallest number that all these divide into is 10.

Step 3: Multiply the Entire Equation by the LCD

Multiply every term in the equation by the LCD of 10 to eliminate the fractions: \[ 10 \left(\frac{1}{10} x^{3}\right) + 10 \left(\frac{1}{2} x^{2}\right) + 10 \left(\frac{1}{5} x\right) - 10 \left(\frac{4}{5}\right) = 0 \]

Step 4: Simplify the Equation

Simplify each term:- \( 10 \times \frac{1}{10} x^{3} = x^{3} \)- \( 10 \times \frac{1}{2} x^{2} = 5x^{2} \)- \( 10 \times \frac{1}{5} x = 2x \)- \( 10 \times \left(-\frac{4}{5}\right) = -8 \)The equation simplifies to: \[ x^3 + 5x^2 + 2x - 8 = 0 \]

Step 5: Verify the Equation

Check to ensure all coefficients are integers. The final equation is \( x^3 + 5x^2 + 2x - 8 = 0 \) and all terms have integer coefficients. This confirms that the multiplication by 10 was applied correctly.

Key Concepts

Multiplication Property of Equality

The multiplication property of equality is a fundamental concept in algebra that maintains the balance of an equation. The idea is simple: if you multiply both sides of an equation by the same number, the equality remains true. This concept is crucial when working to simplify equations, particularly those involving fractions.

In the exercise, multiplying both sides of the equation by the least common denominator (LCD) helped turn fractional coefficients into whole numbers, making the equation easier to solve.

• This action keeps the integrity of the equation intact while simplifying calculations.
• It's a powerful tool for transforming equations into more manageable forms.

The multiplication property ensures that the transformation you apply is consistent across the entire equation, keeping it equivalent to its original form.

Integral Coefficients

Integral coefficients are coefficients that are whole numbers. Having integral coefficients in an algebraic equation simplifies the solving process. Unlike fractions, whole numbers are straightforward to handle, especially in basic arithmetic operations like addition, subtraction, and multiplication.

In this exercise, the goal was to convert an equation with fractional coefficients into one with integral coefficients by using the multiplication property of equality.

• Integral coefficients make the equation more functional and easier to interpret.
• It's easier to identify solutions for equations with whole numbers.

By removing the fractions, the resulting equation is not only more aesthetically pleasing but also simpler to work with, facilitating clearer and more efficient problem-solving strategies.

Least Common Denominator

The least common denominator (LCD) is the smallest number that is a multiple of each of the denominators in a set of fractions. Finding the LCD is essential when you need to eliminate fractions from an equation, as we did in this exercise.

To determine the LCD, identify the denominators of all fractional terms and find the smallest common multiple. In the given exercise, the denominators were 10, 2, and 5, and their least common denominator was 10. Multiplying the entire equation by the LCD clears the fractions efficiently.

• Using the LCD simplifies the equation conversion process.
• It helps maintain the equation's structure while eliminating complex fraction calculations.

This approach streamlines equations, encouraging easier manipulation and resolution.

Fractional Coefficients

Fractional coefficients can often complicate the process of solving algebraic equations. These coefficients are fractions attached to the variables in an equation, indicating a division that needs resolving before proceeding with further calculations.

In this exercise, identifying the fractional coefficients (\(\frac{1}{10} x^{3}\), \(\frac{1}{2} x^{2}\), \(\frac{1}{5} x\), and \(\-\frac{4}{5}\)) was the first step toward simplifying the equation.

• Fractional coefficients need to be handled carefully to avoid calculation errors.
• An important step is to identify these fractions and plan to convert them into whole numbers whenever possible.

By transforming them through multiplication with the LCD, we achieve a version of the equation that is cleaner and easier to solve, making the entire resolution process much more straightforward.