Problem 47 - Prealgebra and Introductory Algebra

Question

Multiply. $$\frac{x^{2}-2 x-24}{x^{2}-5 x-6} \cdot \frac{x^{2}+5 x+6}{x^{2}+6 x+8}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( \frac{x + 3}{x + 1} \).

Step 1: Factor all polynomials

Factorize four polynomials, \(x^{2}-2 x-24 = (x - 6)(x + 4)\), \(x^{2}-5 x-6 = (x - 6)(x + 1)\), \(x^{2}+5 x+6 = (x + 2)(x + 3)\), and \(x^{2}+6 x+8 = (x + 2)(x + 4)\). So, the expression can be rewritten as \(\frac{(x - 6)(x + 4)}{(x - 6)(x + 1)} \cdot \frac{(x + 2)(x + 3)}{(x + 2)(x + 4)}\).

Step 2: Cancel out common terms

Cancel out similar terms from the numerator and denominator and the expression simplifies to \(\frac{(x + 4)}{(x + 1)} \cdot \frac{(x + 3)}{(x + 4)}\).

Step 3: Perform multiplication

After the cancellation of common terms, the remaining terms need to be multiplied following the rule for multiplying fractions, which is to multiply the numerators together and the denominators together, giving us \(\frac{(x + 4)(x + 3)}{(x + 1)}\).

Step 4: Cancel out common terms again

Again, cancel out similar terms from the numerator and denominator and the expression further simplifies to \(\frac{x + 3}{x + 1}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is akin to finding the original pieces of a puzzle; it's the process of breaking down a polynomial into simpler elements that, when multiplied together, yield the original polynomial. It's often the first step in solving polynomial equations or simplifying complex expressions. For instance, the quadratic polynomial \(x^2-2x-24\) can be factorized into \((x-6)(x+4)\).

When factoring, we look for two numbers that both add up to the coefficient of the middle term and multiply to the constant term. In \(x^2-2x-24\), we need two numbers that add up to -2 and multiply to -24. These numbers are -6 and +4. Consequently, the polynomial can be written as the product of two binomials: \((x-6)(x+4)\). Similarly, this approach can be used to factor all the polynomials in the given exercise.

Understanding how to factor effectively is essential, as it can greatly simplify an expression, making it easier to work with in subsequent steps, such as in simplifying rational expressions or canceling common terms.

Simplifying Rational Expressions

Imagine rational expressions as fractions that, instead of plain numbers, contain polynomials in their numerators and denominators. Simplifying these expressions can sometimes feel like tidying up a cluttered room, aiming to make it look as neat as possible. To do so, we often factor polynomials to find common terms that can be canceled out.

For instance, the expression \(\frac{x^2-2x-24}{x^2-5x-6}\), after factorization, takes the form \(\frac{(x-6)(x+4)}{(x-6)(x+1)}\). Here, just like clearing clutter, we look for any common terms in both the numerator and the denominator that can be eliminated. This is possible because any term divided by itself is equal to one. In this case, \(x-6\) appears on the top and bottom and cancels out, leaving us with a cleaner, more manageable expression. Simplifying rational expressions is pivotal to make complex algebraic equations more accessible and less intimidating.

Canceling Common Terms

Canceling common terms is like cutting redundant steps in a path, making the journey shorter and smoother. In algebra, this concept refers to removing identical factors from the numerator and the denominator of a rational expression. After you've factored polynomials, if the same term appears both on the top and bottom of a fraction, you can cancel them out, as they equate to one.

In the example from our exercise, \(\frac{(x + 4)}{(x + 1)} \cdot \frac{(x + 3)}{(x + 4)}\), the term \((x + 4)\) is present in both the numerator and the denominator, making it possible to cancel it. It's essential to remember, however, that terms can only be canceled if they are separated by multiplication. This doesn't apply for addition or subtraction within the rational expression.

Effective cancellation leads to a simpler expression that is easier to manage and solve, which can be particularly useful when you're dealing with high-degree polynomials or complex rational expressions. Being thorough with this process helps avoid mistakes and ensures that you've simplified the expression as much as possible.