Problem 88 - Intermediate Algebra

Question

Perform the operations and simplify. $$ \begin{aligned} &\text { Let } g(s)=\frac{s^{2}-5 s+6}{s^{2}-10 s+16} \text { and } h(s)=\frac{s^{2}-6 s-16}{s^{2}+2 s}\\\ &\text { Find } g(s) \cdot h(s) \end{aligned} $$

Step-by-Step Solution

Answer

The simplified expression for \( g(s) \cdot h(s) \) is \( \frac{s-3}{s} \).

Step 1: Write the expression for the product

The product of two functions \( g(s) \) and \( h(s) \) is given by the expression \( g(s) \cdot h(s) = \left( \frac{s^2 - 5s + 6}{s^2 - 10s + 16} \right) \cdot \left( \frac{s^2 - 6s - 16}{s^2 + 2s} \right) \). This can be simplified as a single fraction: \( \frac{(s^2 - 5s + 6)(s^2 - 6s - 16)}{(s^2 - 10s + 16)(s^2 + 2s)} \).

Step 2: Factor the polynomials in the numerator

Factor the quadratic expressions in the numerator. For \( s^2 - 5s + 6 \), the factors are \((s - 2)(s - 3)\), and for \( s^2 - 6s - 16 \), the factors are \((s - 8)(s + 2)\). Thus, the numerator becomes \( (s - 2)(s - 3)(s - 8)(s + 2) \).

Step 3: Factor the polynomials in the denominator

Factor the quadratic expressions in the denominator. For \( s^2 - 10s + 16 \), the factors are \((s - 2)(s - 8)\), and for \( s^2 + 2s \), factoring out \( s \) gives \( s(s + 2) \). Thus, the denominator becomes \( (s - 2)(s - 8)s(s + 2) \).

Step 4: Simplify the fraction by cancelling common factors

Cancel the common factors in the numerator and the denominator. The common factors \((s - 2)\), \((s - 8)\), and \((s + 2)\) appear in both the numerator and denominator. After cancellation, we are left with \( \frac{s - 3}{s} \).

Step 5: Final simplified expression

The final simplified expression for \( g(s) \cdot h(s) \) after cancelling the common factors is \( \frac{s - 3}{s} \).

Key Concepts

Understanding Multiplying Rational Expressions

Multiplying rational expressions involves managing fractions composed of polynomial expressions. The key idea is to multiply the numerators together and the denominators together, much like you would with regular fractions:

• Combine the numerators of each fraction to form a new numerator.
• Combine the denominators to form a new denominator.

Let's look at our example with the functions \( g(s) \) and \( h(s) \). To multiply these, we multiply:\[\left( \frac{s^2 - 5s + 6}{s^2 - 10s + 16} \right) \cdot \left( \frac{s^2 - 6s - 16}{s^2 + 2s} \right)\]into a single fraction:\[\frac{(s^2 - 5s + 6)(s^2 - 6s - 16)}{(s^2 - 10s + 16)(s^2 + 2s)}\]It's critical to handle each polynomial separately, keeping all terms together but being mindful of the operations and signs.After multiplying, the next step is simplifying the resulting expression, which is often necessary due to complex polynomials in either the numerator or the denominator.

Simplifying Complex Fractions

Once you have multiplied the rational expressions, the next essential step is simplifying the complex fraction. This process includes factoring and canceling out common factors in the numerator and the denominator. Complex fractions can be intimidating, but simplifying them follows a straightforward process:

• Factor each polynomial in the expression. Look for products of factors that are easy to identify, often using methods like grouping or the quadratic formula where applicable.
• Identify and cancel out any common factors shared by the numerator and denominator.

In our exercise with \( g(s) \cdot h(s) \), the common factors were \[(s - 2), (s - 8), \text{ and } (s + 2)\] in both the numerator and the denominator. Canceling these factors left us with the simplified form of the expression:\[\frac{s - 3}{s}\] This simplification reveals the simplest form of the expression, capturing the core essence of the original rational product.

Factoring Quadratic Expressions

Factoring quadratic expressions is critical to simplifying rational expressions, especially when multiplying or dividing them. Quadratics are polynomial expressions of the form \( ax^2 + bx + c \). The goal in factoring is to rewrite these polynomials as a product of simpler binomials if possible. Here's how to factor quadratics like those in our problem:

• Identify two numbers that multiply to \( c \) (the constant term) and add up to \( b \) (the coefficient of the linear term \( x \)). These numbers will be used to break down the expression.
• Rewrite the quadratic in its factored form using these numbers.

For example, with the expression \( s^2 - 5s + 6 \), the numbers \( -2 \) and \( -3 \) multiply to \( 6 \) and add to \( -5 \), allowing us to rewrite the quadratic as \((s - 2)(s - 3)\). Relative to our exercise, factoring made it possible to simplify \( g(s) \cdot h(s) \) by reducing the complexity of the quadratic components, efficiently transforming them into manageable binomials.