Problem 39 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{(x-5)(x+3)}{3 x-1} \cdot \frac{x(3 x-1)}{(x-5)} $$

Step-by-Step Solution

Answer

The expression simplifies to \(x^2 + 3x\).

Step 1: Identify the Parts of the Expression

The given expression is: \[\frac{(x-5)(x+3)}{3x-1} \cdot \frac{x(3x-1)}{(x-5)}\]It consists of two fractions: the first fraction is \( \frac{(x-5)(x+3)}{3x-1} \), and the second is \( \frac{x(3x-1)}{(x-5)} \). We'll need to multiply these fractions and simplify.

Step 2: Simplify the Numerators and Denominators

Notice that the expression contains a common factor \((x-5)\) in the first fraction's numerator and the second fraction's denominator, and \((3x-1)\) is in both fractions, so they can be canceled out:\[= \frac{1 \cdot (x+3)}{1} \cdot \frac{x \cdot 1}{1}\]This simplifies to \((x + 3) \cdot x\).

Step 3: Multiply to Simplify Further

Multiply \((x + 3)\) by \(x\):\[= x(x + 3) = x^2 + 3x\]This results in the simplified expression \(x^2 + 3x\).

Key Concepts

Simplifying Expressions

Simplifying expressions makes mathematical computations easier and reduces complexity, thereby helping in understanding the problem better. This process involves transforming a mathematical expression into its simplest form while maintaining its original value.
To simplify an expression similar to the one given in the exercise, a few steps are generally followed:

• Identify Common Factors: Look for terms or factors that appear in both the numerator and the denominator. This helps in reducing fractions by canceling out these common elements.
• Perform Arithmetic Operations: This may include addition, subtraction, multiplication, or division. In rational expressions, it’s crucial to perform these operations correctly to maintain equality.
• Combine Like Terms: This step involves adding or subtracting terms that have the same variable raised to the same power.

For example, in the given expression, the common factors \(x-5\) and \(3x-1\) were canceled out, leading to a simplified version of \(x(x+3)\). This reduction not only makes the expression simpler but also easier to work with in further calculations.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, constructed using only addition, subtraction, and multiplication operations. Each separate term within a polynomial is called a 'monomial', and collectively they form what is referred to as a polynomial.
In the simplified part of the given exercise, \(x(x+3)\), both ''x'' and \(x+3\) are terms of a polynomial expression, resulting in \(x^2 + 3x\). Here, \(x^2\) and \(3x\) are monomials.
Important features of polynomials include:

• Degree: The degree of a polynomial is the highest power of the variable. In the expression \(x^2 + 3x\), the degree is 2, because the highest power of ''x'' is \(x^2\).
• Terms: These are the individual components of a polynomial. In \(x^2 + 3x\), ''x'' and \(3x\) are the terms.
• Operations: Polynomials can be added, subtracted, multiplied, or divided, like other algebraic expressions; however, division may result in a non-polynomial rational expression.

Understanding polynomials is fundamental in algebra, as they form the basis for exploring more complex mathematical concepts.

Rational Expressions

A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. Simplifying these expressions involves reducing them into the simplest form without changing their value.
The original exercise features two rational expressions being multiplied: \(\frac{(x-5)(x+3)}{3x-1}\) and \(\frac{x(3x-1)}{(x-5)}\). The first step in simplifying rational expressions is to factor both the numerator and the denominator, allowing any common terms to be canceled out.
Key steps in handling rational expressions include:

• Cancelling Common Factors: This is crucial for simplification. Any term appearing both in the numerator and the denominator can be canceled.
• Multiplicative Inverse: When dividing by a fraction, multiply by its reciprocal instead, further simplifying the expression.
• Recognizing Restrictions: For rational expressions, identify values that would make the denominator zero, as these are not part of the domain.

By applying these techniques to the exercise, the complex-looking algebraic fraction is reduced, simplifying further calculations and interpretations.