Problem 21 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 2. $$ \frac{x^{2}+x-6}{5 x} \cdot \frac{5 x-10}{x+3} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{x}\).

Step 1: Factor the Numerators and Denominators

First, we need to factor each of the polynomials in the numerators and denominators, if possible. - For \(x^2 + x - 6\), we factor as \((x + 3)(x - 2)\). - For \(5x - 10\), we factor out the common factor to get \(5(x - 2)\).- The denominator \(5x\) is already in its simplest factored form.- The denominator \(x + 3\) is already in its simplest form.

Step 2: Rewrite the Expression with Factored Terms

Now we rewrite the original expression using the factored forms derived in Step 1:\[\frac{(x + 3)(x - 2)}{5x} \, \cdot \, \frac{5(x - 2)}{x+3}\]

Step 3: Cancel Common Factors

Next, identify and cancel the common factors in the numerators and denominators:- Cancel \((x + 3)\) from the numerator and denominator.- Cancel \((x - 2)\) from the numerator and denominator.- Cancel the factor of 5 since it appears in both the numerator and denominator.After canceling, the expression simplifies to:\[\frac{1}{x}\]

Step 4: Write the Simplified Expression

The simplified form of the given expression, after canceling all common factors, is:\[\frac{1}{x}\]

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra might seem confusing at first, but breaking it down step by step sure helps us out a lot.
When we work with problems like these, we look for ways to make our expressions shorter or more manageable without changing their value.
First off, we focus on simplifying the terms. This often involves identifying terms in numerators and denominators that can be factored further or cancelled. Keep in mind:

• Look for common factors between terms to reduce them.
• Pay attention to polynomials that can be broken down using factorization techniques.

Once you've factored the terms, it's easier to spot terms both above and below the division line in the fraction that can be cancelled out.
Remember, the aim is to reduce the expression to its simplest form, preserving its original meaning and making it easier to work with.

Polynomial Factorization

Polynomial factorization is like finding the secret building blocks of larger pieces.
This process transforms polynomials into products of smaller polynomials.
Polynomials can often be broken down or factored into simpler components, much like you would factor numbers into prime numbers. Common approaches include:

• Finding two numbers that multiply to the constant term and add to the linear coefficient (for quadratics).
• Using techniques like grouping or applying special formulas, such as the difference of squares.

In our exercise, recognizing that the quadratic polynomial \(x^2 + x - 6\) factors into \((x + 3)(x - 2)\) allows for smoother simplification.
Factorization is key in understanding how an expression can be simplified further down the line. It paves the way for identifying common factors across the numerators and denominators in a fraction, essential for simplifying more complex algebraic expressions.

Multiplication of Fractions

Multiplying fractions might sound tricky, but it adheres to a straightforward rule.
All you do is multiply the numerators with numerators and denominators with denominators.
But, in terms of algebraic fractions, factorization before multiplication can save a lot of effort. Here's why:

• Once factored, you might find common factors in the numerators and denominators.
• Cancelling these common factors before you multiply makes the task shockingly simpler.

In algebra, look for opportunities to cancel out as much as possible before getting into the nitty-gritty of multiplying the terms directly.
In our exercise, by factoring the expressions prior to multiplying, significant reduction of complexity was achieved.
Once the expression is simplified, you can proceed with multiplying whatever remains, yielding a simpler form like \(\frac{1}{x}\), which thoroughly represents the simplified result of the original expression.