Problem 90 - Intermediate Algebra

Question

Perform the operations and simplify. $$ \begin{aligned} &\text { Let } g(x)=\frac{x^{2}-9}{x^{2}-49} \text { and } h(x)=\frac{9 x^{2}+27 x}{3 x+21}\\\ &\text { Find } g(x) \div h(x) \end{aligned} $$

Step-by-Step Solution

Answer

\( g(x) \div h(x) = \frac{x-3}{3x(x-7)} \) with \( x \neq 0, \pm7 \).

Step 1: Write down the division expression

When dividing functions, such as \( \frac{g(x)}{h(x)} \), we need to multiply by the reciprocal of the divisor. Therefore, rewrite the expression as \( g(x) \times \frac{1}{h(x)} \).

Step 2: Factor the numerators and denominators

Factor both \( g(x) \) and \( h(x) \) completely: \[ g(x) = \frac{x^2 - 9}{x^2 - 49} = \frac{(x-3)(x+3)}{(x-7)(x+7)} \] \[ h(x) = \frac{9x^2 + 27x}{3x + 21} = \frac{9x(x + 3)}{3(x+7)} \]

Step 3: Setup the division with factored forms

Insert the factored forms back into the division: \[ g(x) \div h(x) = \frac{(x-3)(x+3)}{(x-7)(x+7)} \times \frac{3(x+7)}{9x(x+3)} \]

Step 4: Simplify by canceling common factors

Cancel out the common factors in the numerators and denominators: 1. Cancel \((x+3)\) from both the numerator and the denominator.2. Cancel \((x+7)\) from both the numerator and the denominator.3. Recognize that 9x can be simplified to 3x after canceling the numeric factor 3 from numerator and denominator.Thus leaving:\[ \frac{x-3}{(x-7) \cdot 3x} \]

Step 5: Combine and Simplify the fraction

Simplify the remaining expression:\[ \frac{1}{3x} \times \frac{x-3}{x-7} = \frac{x-3}{3x(x-7)} \]

Step 6: Conclusion

The simplified expression for \( g(x) \div h(x) \) is: \[ \frac{x-3}{3x(x-7)} \]. Ensure to note any restrictions for values of \( x \), which occur where any denominator is zero: \( x eq 0, \pm7 \).

Key Concepts

Polynomial Division

Polynomial division can seem intimidating, but think of it like long division with numbers. It's a way to divide one polynomial by another, resulting in a quotient and a possible remainder.
When dividing functions, you express the division as multiplication by the reciprocal. Here, to divide \( g(x) \) by \( h(x) \), you need to write it as:

• \( g(x) \times \frac{1}{h(x)} \)

This makes it easier to understand and perform division, converting it into a straightforward multiplication process.
The key steps are:

• Factor each polynomial completely.
• Multiply by the reciprocal of the polynomial divisor.
• Finally, simplify the resulting expression by canceling common factors.

In our specific exercise, identifying common factors in both functions allowed us to rewrite the division in simplified terms.

Factoring Expressions

Factoring is a powerful tool in algebra, especially when working with rational expressions. The main idea is to break down complex expressions into simpler 'factors' that multiply to restore the original expression.
First, recognize patterns like difference of squares. For instance:

• \( x^2 - 9 = (x-3)(x+3) \)
• \( x^2 - 49 = (x-7)(x+7) \)

Also, identify any common factors. In the expression \( 9x^2 + 27x \), factor out \( 9x \), giving:

• \( 9x(x + 3) \)

This method not only simplifies expressions but is also crucial in the simplification process of rational expressions.
By successfully factoring both the numerator and the denominator, we make it easier to spot and cancel common terms, significantly simplifying the division.

Simplifying Expressions

Simplification is about reducing expressions to their most basic and clean form for ease of interpretation and further calculation. To simplify a rational expression fully:

• Identify and cancel common factors in the numerator and the denominator.
• Be cautious to only cancel entire factors, not terms.

In our exercise, once we factor all terms and set up the division correctly, we look for common factors. These factors appear in both products. By correctly canceling:

• The \((x+3)\) and \((x+7)\) terms are cancelled.
• The \(3\) from the constant term is used to reduce the fraction, simplifying \( \frac{9x}{3} \) to \(3x\).

The end result \( \frac{x-3}{3x(x-7)} \) reflects this simplification.
However, remember to consider domain restrictions. Any values of \( x \) that make the denominator zero should be avoided to keep the expression valid.