Problem 8 - A Graphical Approach to College Algebra

Question

Each expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$\frac{7 x^{8}-6 x^{3}}{6 x^{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{7x^6}{6} - x \).

Step 1: Identify the structure of the expression

The given expression is \( \frac{7x^8 - 6x^3}{6x^2} \). This expression can be split into two separate fractions using the property: \( \frac{a+b}{c} = \frac{a}{c} + \frac{b}{c} \), which will help us to simplify the expression.

Step 2: Apply the property to split the fraction

Use the property \( \frac{a+b}{c} = \frac{a}{c} + \frac{b}{c} \) on the expression \( \frac{7x^8 - 6x^3}{6x^2} \). This results in two separate fractions: \( \frac{7x^8}{6x^2} - \frac{6x^3}{6x^2} \).

Step 3: Simplify the first fraction

Take the first fraction \( \frac{7x^8}{6x^2} \). Divide both the numerator and the denominator by \( x^2 \), which simplifies to \( \frac{7x^{8-2}}{6} = \frac{7x^6}{6} \).

Step 4: Simplify the second fraction

Now simplify the second fraction \( \frac{6x^3}{6x^2} \). Divide the numerator and the denominator by \( x^2 \), resulting in \( \frac{6x^{3-2}}{6} = \frac{6x}{6} \), and then simplify it further to \( x \).

Step 5: Combine the simplified fractions

Combine the simplified fractions from Steps 3 and 4: \( \frac{7x^6}{6} - x \). This gives us the final simplified expression.

Key Concepts

Fraction Decomposition

Fraction decomposition is a technique used to split a complex fraction into simpler fractions. This method helps in breaking down an expression for easier manipulation and calculation. In the original exercise, the expression given is \( \frac{7x^8 - 6x^3}{6x^2} \). To decompose this fraction, we use the property: \( \frac{a+b}{c} = \frac{a}{c} + \frac{b}{c} \).

This property allows division to be applied to each term in the numerator individually. Substituting the values from the exercise gives us two separate fractions: \( \frac{7x^8}{6x^2} \) and \( \frac{-6x^3}{6x^2} \).

• This simplifies the problem significantly because each term can now be analyzed independently.
• Remember to always check if a common factor exists to simplify expressions further.

By decomposing the fraction, we can focus on simplifying each part individually, making the entire simplification process more straightforward.

Polynomial Division

Polynomial division is a method distinct from regular numerical division but functions similarly to long division. When dividing polynomials, especially within fractions, you extract common factors from each term individually. In our context, polynomial division is used to simplify each fraction after decomposition.

For the first fraction, \( \frac{7x^8}{6x^2} \), divide both the numerator and the denominator by \( x^2 \). This simplifies the expression to \( \frac{7x^{8-2}}{6} \), resulting in \( \frac{7x^6}{6} \).

Similarly for the second fraction, \( \frac{-6x^3}{6x^2} \), you also divide the numerator and denominator by \( x^2 \), leading to \( \frac{-6x^{3-2}}{6} = \frac{-6x}{6} \), or simply \(-x\).

• Always focus on canceling out the highest common factor.
• Ensure exponents are adjusted according to division rules.

Thus, polynomial division greatly aids in simplifying expressions into more manageable terms.

Exponent Rules

Exponent rules are crucial when simplifying algebraic expressions. They allow easy manipulation of powers, making expressions easy to handle. One key rule is that when dividing like bases, you subtract the exponents: \( \frac{x^a}{x^b} = x^{a-b} \).

In the application of exponent rules from the exercise, observe how the rule is used for both fractions. Initially, \( \frac{7x^8}{6x^2} \) becomes \( \frac{7x^{8-2}}{6} = \frac{7x^6}{6} \). Likewise, \( \frac{6x^3}{6x^2} \) turns into \( 6x^{3-2} \) divided by 6, simplifying to \( x \).

• This rule streamlines simplifying powers of variables.
• It ensures you maintain control over the entire expression's form.

Understanding and applying these rules correctly ensures expressions are simplified efficiently, reducing errors in algebraic manipulation.