Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Simplify each rational 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

\(\frac{7}{6}x^6 - x\)

Step 1: Factor out the Greatest Common Divisor (GCD)

Identify the greatest common factor from the numerator. In the expression \(7x^8 - 6x^3\), the GCD is \(x^3\). Factor this from the numerator to get \(x^3(7x^5 - 6)\).

Step 2: Split the Rational Expression

Apply the property \(\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}\) to separate the terms in the numerator: \(\frac{7x^8}{6x^2} - \frac{6x^3}{6x^2}\).

Step 3: Simplify Each Term

Simplify the separated terms individually. For \(\frac{7x^8}{6x^2}\), divide the coefficients and subtract the exponents for \(x\): \(\frac{7}{6}x^{8-2} = \frac{7}{6}x^6\). For \(\frac{6x^3}{6x^2}\), divide the coefficients and subtract the exponents: \(x^{3-2} = x\).

Step 4: Combine the Simplified Expression

Combine the simplified terms: \(\frac{7}{6}x^6 - x\).

Key Concepts

Greatest Common Divisor (GCD)

The concept of the Greatest Common Divisor (GCD) is crucial when simplifying rational expressions. The GCD refers to the largest factor that divides two or more numbers or terms without leaving a remainder. In the context of polynomial expressions, the GCD is used to factor out common terms.

For an expression like \(7x^8 - 6x^3\), identifying the GCD involves breaking down each term into its prime factors. Here, each term includes the variable \(x\). To find the GCD, look for the highest power of \(x\) that appears in every term, which in this case is \(x^3\). By factoring \(x^3\) out of the entire expression, we transform it into \(x^3(7x^5 - 6)\).

This step significantly simplifies the expression and prepares it for further operations.

Simplifying Expressions

Simplifying rational expressions involves reducing the expression to its simplest form. After factoring out the GCD in a rational expression, the next step often includes simplifying further by splitting the terms in the expression.

In our exercise, once \(x^3\) is factored out, we can apply the property \(\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}\). This allows us to look at each term independently. For example, \(\frac{7x^8 - 6x^3}{6x^2}\) can be split into two separate fractions: \(\frac{7x^8}{6x^2} - \frac{6x^3}{6x^2}\).

Each of these fractions can be further simplified by dividing coefficients and subtracting the exponents, which paves the way for the final simplified expression. Simplifying expressions in this manner helps in better understanding and working with complex algebraic fractions.

Polynomial Division

Polynomial division is an integral part of simplifying rational expressions. After splitting the expression into separate fractions, the process typically involves polynomial division, i.e., dividing the terms in the numerator by the terms in the denominator.

To illustrate, let's consider \(\frac{7x^8}{6x^2}\). Here, polynomial division involves dividing the coefficient 7 by 6, then handling the variable part by subtracting the exponent of the denominator from that of the numerator: \(x^{8-2} = x^6\). This results in \(\frac{7}{6}x^6\).

Polynomial division isn't just about numerical coefficients; it involves simplifying powers of variables to bring an often complex expression into a simpler, more digestible form. This is highly beneficial for clear analysis and application in broader mathematical problems.