Problem 30 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 3. $$ \frac{9 x^{2}+3 x-20}{3 x^{2}-7 x+4} \cdot \frac{3 x^{2}-5 x+2}{9 x^{2}+18 x+5} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{3x-2}{3x+1}\).

Step 1: Factor Numerators and Denominators

First, let's factor the numerators and denominators of both fractions. For the first fraction, the numerator is \(9x^2 + 3x - 20\). This can be factored into \((3x-4)(3x+5)\). The denominator \(3x^2 - 7x + 4\) factors into \((3x-4)(x-1)\). For the second fraction, the numerator \(3x^2 - 5x + 2\) can be factored into \((3x-2)(x-1)\) and the denominator \(9x^2 + 18x + 5\) into \((3x+5)(3x+1)\).

Step 2: Multiply the Factored Expressions

Multiply the factored forms of the numerators and the denominators. Numerator: \((3x-4)(3x+5) \cdot (3x-2)(x-1)\)Denominator:\((3x-4)(x-1) \cdot (3x+5)(3x+1)\)

Step 3: Cancel Common Factors

Since the expression is a multiplication of fractions, we can cancel out the common factors in the numerators and denominators. Cancel \((3x-4)\) and \((3x+5)\) which appear in both the numerator and denominator. Also, \((x-1)\) is a common factor, so we cancel that as well. We're left with the following: Numerator: \((3x-2)\)Denominator: \((3x+1)\)

Step 4: Write the Simplified Expression

After canceling the common factors, the simplified expression is left as:\[\frac{3x-2}{3x+1}\]. This fraction is already simplified as there are no further common factors.

Key Concepts

Factoring Polynomials

When dealing with polynomials, factoring plays a crucial role. Factoring involves expressing a polynomial as a product of its smaller polynomials, or factors, which can simplify the process of solving equations or working with expressions.
For instance, take the polynomial \(9x^2 + 3x - 20\). To factor this, look for two numbers that multiply to give the first coefficient multiplied by the last coefficient (\(9 \times -20 = -180\)) and add up to the middle coefficient (3). After finding such numbers, rewrite the middle term and group the terms to factor by grouping. Hence, \(9x^2 + 3x - 20\) factors into \((3x-4)(3x+5)\).
Understanding factoring helps in working with more complex algebraic expressions, as it allows for simplification and easier manipulation during mathematical operations.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Much like numerical fractions, these expressions can be simplified by factoring and canceling common factors.
To work with rational expressions effectively, remember:

• Factor both the numerator and denominator completely.
• Identify any common factors.
• Cancel the common factors to simplify the expression.

In our given problem, starting with rational expressions such as \(\frac{9x^2 + 3x - 20}{3x^2 - 7x + 4}\), facilitates the process of multiplying fractions by allowing us to focus only on the relevant components of the expression. Rational expressions offer a clean and straightforward way to perform operations with algebraic fractions once you've grasped the basics of factoring.

Multiplying Fractions

Multiplying fractions involves a straightforward process: multiply the numerators with each other and the denominators with each other. This same principle applies to rational expressions. However, before multiplying, it's often beneficial to factor the numerators and denominators, as seen in the original exercise.
Once factored, multiply the factored expressions carefully to maintain accuracy in your calculations. With our example, you would multiply \((3x-4)(3x+5)\) with \((3x-2)(x-1)\) for the numerator. Then, the denominators \((3x-4)(x-1)\) and \((3x+5)(3x+1)\) are multiplied together.
Understanding multiplication of fractions not only aids in managing complexity but also prepares you for tackling larger algebraic problems by building a foundation of strong multiplication skills.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of reducing expressions to their most concise form. This involves combining like terms, canceling common factors, and reducing overall complexity.
In the exercise we worked on, simplification occurred after the multiplication of factored forms. Cancel out common factors from both the numerator and denominator. For our expression, \((3x-4)\), \((3x+5)\), and \((x-1)\) were common factors that could be canceled, leading to a simpler expression \(\frac{3x-2}{3x+1}\).

• Always check for any remaining factors that can be canceled.
• Ensure no common factors remain so that the expression is fully simplified.

Knowing how to simplify helps keep work organized, reduces errors, and makes understanding the core problem much easier. Simplification is a fundamental tool in algebra that brings clarity and order to complex workings.