Problem 24 - Elementary Algebra

Question

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{1}{z^{2}+9 z+14}}{z^{2}-49} $$

Step-by-Step Solution

Answer

z is an unknown value in an indefinitely length of the common factor which divides. Factorize the common denominator of division into: z+2. The simplified fraction would be above.

Step 1: Simplify the complex fraction

\(\frac{\frac{1}{z^2+9z+14}}{z^2-49} = \frac{1}{(z^2+9z+14)(z^2-49)}\). Factor: \(z^2+9z+14 = (z+2)(z+7)\) and \(z^2-49 = (z-7)(z+7)\).

Step 2: Simplify

\(= \frac{1}{(z+2)(z+7)(z-7)(z+7)} = \frac{1}{(z+2)(z+7)^2(z-7)}\)

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra. It involves rewriting a polynomial as a product of its factors. This makes complex expressions simpler and is important in solving polynomial equations. Take for example, the expression: \( z^2 + 9z + 14 \). To factor this, you need to find two numbers that multiply to give 14 (the constant term) and add to give 9 (the coefficient of the middle term, z). These two numbers are 2 and 7. So, \( z^2 + 9z + 14 = (z + 2)(z + 7) \). By factoring, we can simplify the polynomial and use it in further calculations, as seen later in rational expressions.

Rational Expressions

A rational expression is a fraction where the numerator and/or the denominator are polynomials. For example: \( \frac{1}{z^2 + 9z + 14} \) is a rational expression. Simplifying these expressions often requires factoring the polynomials in the numerator and the denominator. In our exercise, we first factored the numerator: \( \frac{1}{(z+2)(z+7)} \). Then, we also need to examine the denominator, which in our case is another polynomial \( z^2 - 49 \). Factoring these parts helps make the overall expression simpler and more manageable.

Algebraic Fractions

Algebraic fractions, also known as rational expressions, involve fractions with polynomial expressions in the numerator and/or the denominator. Simplifying these fractions takes a few steps: identify any common factors, factor the polynomials, and then reduce if possible. Consider: \( \frac{\frac{1}{z^2 + 9z + 14}}{z^2 - 49} \). First, factor both the numerator and the denominator. For the numerator, we already found: \( \frac{1}{(z+2)(z+7)} \). For the denominator: \( z^2 - 49 = (z+7)(z-7) \). Now, placing these factored forms into the fraction, it becomes: \( \frac{1}{(z+2)(z+7)} \times \frac{1}{(z-7)} \).

Common Factors

Identifying common factors is crucial for simplifying rational expressions. This involves spotting terms that appear in both the numerator and the denominator of a fraction. Once we factor both parts—like \( \frac{1}{(z+2)(z+7)} \times \frac{1}{(z-7)} \) — we look for common factors. In our example, the terms \( (z+7) \) appear in both the numerator and one part of the denominator, and they can be canceled out. This results in: \( \frac{1}{(z+2)(z-7)} \). Thus, simplifying by cancelling common factors helps to make algebraic fractions much easier to work with and to solve.