Problem 90 - A Graphical Approach to Precalculus with Limits

Question

Divide. $$\left(-x^{2}-1\right) \div(3 x-9)$$

Step-by-Step Solution

Answer

The expression is \(\frac{-x^2 - 1}{3(x-3)}\), as it cannot be simplified further.

Step 1: Identify the Division Expression

We need to divide the polynomial \(-x^{2}-1\) by the linear expression \(3x-9\). The expression to solve is: \(\frac{-x^2-1}{3x-9}\).

Step 2: Factor the Denominator

The denominator \(3x - 9\) can be factored. By taking out the common factor of 3, it becomes \(3(x - 3)\).

Step 3: Set Up Polynomial Long Division

Set up the polynomial long division with dividend \(-x^2 - 1\) and divisor \(3(x-3)\).

Step 4: Perform Long Division

Since the highest degree in the numerator \(-x^2 - 1\) does not allow it to be divided directly by \(3x\), we conclude \(-x^2 - 1\) cannot be simplified by division. Thus, the expression is left in its rational form.

Step 5: Express in Simplified Form (If Possible)

Since the terms do not simplify (as discussed in Step 4), the division results in the expression itself, kept in the rational form: \(\frac{-x^2 - 1}{3(x-3)}\).

Key Concepts

Polynomial Long Division

Polynomial Long Division is a method we use to divide one polynomial by another. Much like dividing numbers, this process involves taking the first term of the dividend and dividing it by the first term of the divisor. The result is then multiplied by the entire divisor and subtracted from the dividend, creating a new polynomial. This process repeats until we've reduced the problem as much as possible. Here's how it works step-by-step:

• Write the dividend and divisor in standard form, which means arranging the terms in descending order of their degrees.
• Divide the leading term of the dividend by the leading term of the divisor. This gives the first term of the quotient.
• Multiply the entire divisor by this term and subtract the result from the dividend.
• Repeat this process with the new polynomial you get after subtraction, continuing until the degree of the remaining polynomial is less than the degree of the divisor.

Unlike numbers, polynomials don't always divide neatly, which means there might be a remainder. This remainder becomes part of the final expression. In this exercise, since the division cannot be performed straight away due to the higher degree of the numerator, the expression remains as a rational fraction.

Rational Expressions

A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. These expressions can often be simplified or rewritten through factoring and division.Working with rational expressions involves:

• Identifying common factors in the numerator and denominator and canceling them out, provided that the values they represent do not create undefined expressions (like division by zero).
• Understanding the restrictions on the variable, which often arise from the denominator. Denominators should never equal zero, so identifying these restrictions is crucial.

In the context of the given exercise, \[ \frac{-x^2 - 1}{3x - 9} \]both the numerator \(-x^2 - 1\), and the denominator, \(3x - 9\), were put under scrutiny to find simplifications. The first step was factoring the denominator down to \(3(x - 3)\), but since nothing matches from the numerator, the form is kept as it is. Rational expressions are key in algebra, helping us understand underlying relationships without having to solve complex equations.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of simpler polynomials. It's akin to breaking down a larger object into smaller, more manageable parts. This is useful not just in simplifying expressions but also in solving equations and understanding polynomial behavior.The key steps in factoring include:

• Finding the greatest common factor (GCF) of the terms.
• Rewriting the polynomial as a product of its factors, which allows for simpler manipulation.
• Checking if the result can be factored further, especially in cases of quadratic polynomials, which can often be broken down into binomials.

In our provided exercise, the polynomial in the denominator, \(3x - 9\), was factored out using the GCF, which is 3, to become \(3(x - 3)\). Factoring makes it clear whether a polynomial can be neatly divided or simplified further. It is an essential skill for simplifying rational expressions and performing polynomial division effectively.