Problem 27 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\(\frac{5}{x}\)

Step 1: Factoring the Expressions

To simplify the multiplication of two fractions, start by factoring each part. The numerator of the first fraction is a quadratic expression: \(x^2 + x - 6\). We can factor it into \((x + 3)(x - 2)\).The numerator of the second fraction is a linear expression: \(5x - 10\). Factor out the common term, which is 5, to get \(5(x - 2)\).

Step 2: Rewrite the Expression with Factors

Rewriting the fractions with their factors helps to simplify:\[ \frac{(x+3)(x-2)}{5x} \cdot \frac{5(x-2)}{x+3} \]

Step 3: Cancel Common Factors

To simplify the multiplication of these fractions, cancel out the common factors from the numerators and denominators:- \((x + 3)\) appears in the numerator of the first fraction and the denominator of the second fraction.- \((x - 2)\) appears in the numerators of both fractions.After canceling these common factors, the expression becomes:\[ \frac{1}{x} cdot \frac{5}{1} \]

Step 4: Multiply the Simplified Terms

Multiply the simplified terms:\[ \frac{1 \cdot 5}{x \cdot 1} = \frac{5}{x} \]

Step 5: Final Expression

After performing the multiplication and simplification, the final expression is:\[ \frac{5}{x} \]. This is already in its simplest form since there are no further factors to cancel.

Key Concepts

Factoring: Breaking Down Expressions

Factoring is akin to breaking down a problem into smaller, more manageable pieces. It involves expressing a polynomial as a product of its factors. This process makes complex algebraic manipulations simpler and can reveal structures that aren't immediately apparent.

For instance, consider the quadratic expression:

• \(x^2 + x - 6\): This expression can be factored into \((x + 3)(x - 2)\). Here, we're looking for two numbers that multiply to \(-6\) and add to \(1\), the coefficient of \(x\).
• \(5x - 10\): This is a linear expression that can be factored by taking out the common factor of 5, resulting in \(5(x - 2)\).

Breaking down expressions in this way is useful not only for simplifying algebraic fractions but also for solving equations and finding roots. Each factor represents a potential point of intersection when graphed or a solution to the equation.

Simplifying Expressions: Making Math Manageable

Simplifying an expression involves reducing it to the most concise form possible. In the context of algebraic fractions, simplification often means canceling out common factors between numerators and denominators. This process makes calculations easier and often provides insights into the problem.

After factoring, our expressions in the problem become:

• \( \frac{(x+3)(x-2)}{5x} \cdot \frac{5(x-2)}{x+3} \)

We can cancel the common terms. The common factors \((x + 3)\) and \((x - 2)\) appear in both numerators and denominators and thus can be canceled. This cancellation is a crucial step to ensure the resulting fraction is in its simplest form before any multiplication.

Multiplying Fractions: Combining Simplified Terms

Multiplying fractions involves a straightforward process where we multiply the numerators together and the denominators together. However, before doing this, simplifying each fraction can save time and effort by reducing the complexity. Once the fractions are simplified, the multiplication becomes highly efficient.

Following simplification, our expression transforms to:

• \( \frac{1}{x} \cdot \frac{5}{1} \)

Now, the multiplication simply consists of:

• \( 1 \times 5 \) for the numerator, and
• \( x \times 1 \) for the denominator.

Which results in:

• \( \frac{5}{x} \)

This fraction is already simplified. Multiplying fractions after simplifying ensures we achieve the most reduced version of the product, which makes it easier to understand and use in subsequent calculations.