Problem 29 - A Graphical Approach to College Algebra

Question

Find each product or quotient. $$\frac{p^{2}-p-12}{p^{2}-2 p-15} \cdot \frac{p^{2}-9 p+20}{p^{2}-8 p+16}$$

Step-by-Step Solution

Answer

\(\frac{1}{p-4}\)

Step 1: Factor the Numerators

First, let's factor the numerators of the given expression. The first numerator, \(p^2 - p - 12\), can be factored into \((p-4)(p+3)\). For the second numerator, \(p^2 - 9p + 20\), it factors into \((p-5)(p-4)\).

Step 2: Factor the Denominators

Next, factor the denominators. The first denominator, \(p^2 - 2p - 15\), factors to \((p-5)(p+3)\). The second denominator, \(p^2 - 8p + 16\), is a perfect square trinomial and factors to \((p-4)^2\).

Step 3: Simplify the Expression by Cancelling Common Factors

Let's rewrite the expression with the factors and cancel any common factors from the numerators and denominators. The expression becomes:\[\frac{(p-4)(p+3)}{(p-5)(p+3)} \, \cdot \, \frac{(p-5)(p-4)}{(p-4)^2}\]The common factor between the numerators and denominators is \((p-4)\) in both fractions, \((p+3)\), and \((p-5)\). Cancelling these common factors, we are left with:\[\frac{1}{(p-4)}\]

Step 4: Final Simplification

After cancelling the common factors, the simplified expression is \(\frac{1}{p-4}\), as all other elements have cancelled out.

Key Concepts

Factoring Polynomials

To solve problems involving algebraic fractions, understanding how to factor polynomials is very important. Factoring means writing a polynomial as a product of polynomials of lower degree. This process helps in simplifying the expression and finding common factors.

When factoring a polynomial like the quadratic expression, you begin by identifying patterns such as the difference of squares, perfect square trinomials, or simply factoring by grouping. Patterns make it easier and quicker to factor. For instance, our example uses the quadratic expressions:

• The polynomial \( p^2 - p - 12 \) can be factored into \((p-4)(p+3)\). This is done by finding two numbers that multiply to -12 and add to -1.
• Similarly, \( p^2 - 9p + 20 \) becomes \((p-5)(p-4)\), based on two numbers multiplying to 20 and adding to -9.
• A perfect square trinomial like \( p^2 - 8p + 16 \) factors to \((p-4)^2\). This is recognized because it fits the pattern \( (a-b)^2 = a^2 - 2ab + b^2 \).

Recognizing these patterns is crucial as it lays the groundwork for simplifying rational expressions by canceling out common terms.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing the expression to its simplest form. This process includes canceling out common factors that appear in both the numerator and the denominator.

In our example, once we have factored the expressions, they look like this:

• Numerator: \((p-4)(p+3)\) and \((p-5)(p-4)\)
• Denominator: \((p-5)(p+3)\) and \((p-4)^2\)

Each factor appears multiple times, providing an opportunity to cancel them. It's key to only cancel common factors across numerators and denominators.

Specifically here, we cancel

• \((p-4)\) from all relevant locations
• \((p+3)\)
• \((p-5)\)

Once cancellation is complete, you're left with whatever factors remain, simplifying our example to \( \frac{1}{p-4} \). Simplification allows us to manage expressions more easily, and where applicable, solve equations or evaluate expressions with greater ease.

Multiplying Fractions

Multiplying fractions involves a straightforward pattern: the numerators are multiplied together, as are the denominators. Unlike addition or subtraction of fractions, there is no need for a common denominator.

When multiplying algebraic fractions, like in our example, it is beneficial to factor first. This step increases the likelihood of identifying common factors that can be cancelled, simplifying the final expression.

Consider our expression:

• \( \frac{p^2 - p - 12}{p^2 - 2p - 15} \cdot \frac{p^2 - 9p + 20}{p^2 - 8p + 16} \)

After factoring, we have:

• Numerators: \((p-4)(p+3)\) and \((p-5)(p-4)\)
• Denominators: \((p-5)(p+3)\) and \((p-4)^2\)

Once factors are in place, multiplication proceeds with new, reduced fractions:

• Multiply across the numerators: resulting in 1, after cancellation
• Multiply across the denominators: resulting in \(p-4\), after cancellation

Following these steps ensures accuracy in obtaining the final product, which balances simplicity and precision, giving us \( \frac{1}{p-4} \). This method applies to any pair of fractions and is a crucial skill in algebra.