Problem 22 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 2. $$ \frac{z^{2}+4 z-5}{25 z-25} \cdot \frac{5 z}{z+5} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{z}{5}\).

Step 1: Factor the Quadratic Expression

The expression \( z^{2} + 4z - 5 \) can be factored into \((z + 5)(z - 1)\). This is obtained by finding two numbers that multiply to \(-5\) and add to \(4\), which are \(5\) and \(-1\).

Step 2: Factor the Denominator

Factor \(25z - 25\) by taking out the common factor, which is 25. So, \(25z - 25 = 25(z - 1)\).

Step 3: Arrange the Expressions

Reorganize the original expression as follows:\[\frac{(z+5)(z-1)}{25(z-1)} \times \frac{5z}{z+5}\]

Step 4: Cancel Common Factors

Identify and cancel common factors in the numerator and denominator:- Cancel \(z+5\) from both the numerator of the first fraction and the denominator of the second fraction.- Cancel \(z-1\) from both the numerator and the denominator of the first fraction.

Step 5: Simplify the Expression

After canceling, the expression simplifies to:\[\frac{1}{25} \times 5z\]which further simplifies to \(\frac{5z}{25}\).

Step 6: Final Simplification

Simplify \(\frac{5z}{25}\) by cancelling common factors in the numerator and the denominator:- Divide both by 5 to get \(\frac{z}{5}\).

Key Concepts

Factoring Quadratics

Factoring quadratics involves breaking down a quadratic expression, like \(z^2 + 4z - 5\), into two simpler expressions that multiply together. The goal is to find two numbers that not only multiply to the constant term \(-5\) but also add up to the coefficient of the middle term, which is \(4\). This is a fundamental principle of factoring quadratics.

• In the given expression, those two numbers are \(5\) and \(-1\).
• Therefore, the expression \(z^2 + 4z - 5\) can be rewritten as \((z + 5)(z - 1)\).

This method is handy because it simplifies expressions, making them easier to manipulate later in calculations. Factoring is essential when working to simplify complex algebraic expressions, especially when multiplying or dividing rational expressions.

Rational Expressions

Rational expressions are fractions that have polynomials in the numerator, the denominator, or both. Understanding rational expressions means knowing how to manipulate these fractions to simplify them as much as possible.
In the provided exercise, each fraction is a rational expression. We have:

• Nominator: \((z + 5)(z - 1)\) and \(5z\)
• Denominator: \(25(z - 1)\) and \(z + 5\)

To simplify, we aim to cancel out common factors from the numerator and the denominator wherever possible.

• For instance, \(z + 5\) appears in both the numerator and the denominator, and we can cancel it out.
• Similarly, cancel \(z - 1\) which is present in both top and bottom of the rational expressions.

Once common factors are eliminated, you only perform operations with the simplified elements, making calculations more manageable.

Multiplication of Fractions

Multiplying fractions involves multiplying the numerators together and the denominators together. The key to making this process efficient is to simplify each fraction as much as possible before doing any multiplication.
In our exercise, after factoring and canceling, the expression becomes a simple matter of multiplication:

• The simplified expression is \(\frac{1}{25} \times 5z\).
• This multiplication yields \(\frac{5z}{25}\).

Multiplication here is straightforward because each part of the expression has already been simplified. The last step is simplifying \(\frac{5z}{25}\) by canceling common factors:

• Both the numerator and the denominator are divisible by 5.
• Dividing them gives \(\frac{z}{5}\).

The multiplication of fractions results in a neat, approachable expression that can be easily used in further calculations or applications.