Problem 97 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{z^{2}-9}{z^{2}+4 z+3} \div \frac{z^{2}-3 z}{(z+1)^{2}}$$

Step-by-Step Solution

Answer

The simplified result is \( \frac{z+1}{z} \).

Step 1: Factorize the Numerator and Denominator of the First Fraction

The expression is \( \frac{z^2 - 9}{z^2 + 4z + 3} \). Begin by factoring both the numerator and the denominator. The numerator \( z^2 - 9 \) is a difference of squares: \( z^2 - 9 = (z - 3)(z + 3) \). The denominator \( z^2 + 4z + 3 \) is factored into \((z + 1)(z + 3) \). Thus, the first fraction becomes: \( \frac{(z-3)(z+3)}{(z+1)(z+3)} \).

Step 2: Factorize the Numerator and Denominator of the Second Fraction

The second fraction is \( \frac{z^2 - 3z}{(z+1)^2} \). Factor the numerator \( z^2 - 3z \) by taking out the common factor \( z \): \( z(z - 3) \). The denominator is already in its simplest form: \( (z+1)^2 \). Therefore, the second fraction becomes: \( \frac{z(z-3)}{(z+1)^2} \).

Step 3: Rewrite the Division as a Multiplication

To divide by a fraction, multiply by its reciprocal. This changes our problem from division to multiplication:\[ \frac{(z-3)(z+3)}{(z+1)(z+3)} \times \frac{(z+1)^2}{z(z-3)} \].

Step 4: Simplify the Product by Canceling Common Factors

Before multiplying, cancel common factors from numerators and denominators. - Cancel \( (z-3) \) from both a numerator and a denominator.- Cancel \( (z+3) \) since it appears in both a numerator and a denominator. This simplifies the expression to: \[ \frac{(z+1)}{z} \].

Step 5: Final Simplification

Now, the only remaining expression is \( \frac{z+1}{z} \). The result is already simplified.

Key Concepts

Factoring

Factoring is the process of breaking down complex expressions into simpler components, called factors, that when multiplied together give the original expression. This is a key step in solving algebraic problems, especially with rational expressions.
Let's focus on the example provided:

• For the numerator of the first fraction, which is \(z^2 - 9\), notice it is a difference of squares. A difference of squares can be factored as \((a^2 - b^2) = (a-b)(a+b)\). Here, \(z^2 - 9\) becomes \((z - 3)(z + 3)\).
• The denominator \(z^2 + 4z + 3\) can be factored into two binomial expressions, which is \((z + 1)(z + 3)\). Identify the middle term and constants that multiply to 3 and add to 4.
• For the second fraction \(z^2 - 3z\), take out the greatest common factor (GCF), which in this case is \(z\), resulting in \(z(z - 3)\).

Factoring helps to simplify rational expressions and facilitates operations like multiplication and division of these expressions.

Multiplication of Rational Expressions

When you multiply rational expressions, much like multiplying fractions, you multiply the numerators together and the denominators together. Let’s examine this step using our provided problem.
Before starting the multiplication, rewrite the division as multiplication by taking the reciprocal of the second fraction:

• The problem \(\frac{z^2-9}{z^2+4z+3} \div \frac{z^2-3z}{(z+1)^2}\) becomes \(\frac{z^2-9}{z^2+4z+3} \times \frac{(z+1)^2}{z^2-3z}\).

Multiply:

• Numerators: \((z - 3)(z + 3)\times (z+1)^2.\)
• Denominators: \((z+1)(z+3)\times z(z-3).\)

In multiplication, keep an eye out for common factors which can be cancelled to simplify the expression. It leads directly into the simplification step, where you can remove identical factors from both tops and bottoms before performing the actual multiplication, thus simplifying significantly early in the process.

Division of Rational Expressions

Division of rational expressions involves converting division into multiplication by using the reciprocal of the divisor. This step is crucial to transform complex division operations into more straightforward multiplication tasks.
Here’s how it works with the example:

• The original division problem is \(\frac{z^2-9}{z^2+4z+3} \div \frac{z^2-3z}{(z+1)^2}\).
• By rewriting it as multiplication, the second fraction becomes its reciprocal, \(\frac{(z+1)^2}{z(z-3)}\).

Now, multiply as we outlined in the earlier multiplication section.

The beauty of this approach lies in its ability to streamline the process. Division can be trickier than multiplication, and this conversion simplifies it. Remember always to simplify before multiplying after converting division to multiplication, as this lets you cancel out common factors, avoiding unnecessary complications and reducing your workload significantly.