Problem 23 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 2. $$ \frac{x^{2}+2 x+1}{9 x^{3}} \cdot \frac{2 x^{2}-2 x}{2 x^{2}-2} $$

Step-by-Step Solution

Answer

The simplified product is \( \frac{x+1}{9x^{2}} \).

Step 1: Simplify the right fraction

First, look at the fraction on the right: \( \frac{2x^{2}-2x}{2x^{2}-2} \). Notice that both the numerator and the denominator can be factored. The numerator \( 2x^{2}-2x \) can be factored as \( 2x(x-1) \), while the denominator \( 2x^{2}-2 \) can be factored as \( 2(x^{2}-1) \) or \( 2(x-1)(x+1) \) since \( x^{2}-1 \) is a difference of squares. After canceling the common factor \( x-1 \), we get: \( \frac{2x}{2(x+1)} \) which simplifies to \( \frac{x}{x+1} \).

Step 2: Multiply the fractions

Now, multiply the simplified right fraction with the left fraction: \[ \frac{x^{2}+2x+1}{9x^{3}} \cdot \frac{x}{x+1} \] You can write it as a single fraction: \[ \frac{(x^{2}+2x+1)x}{9x^{3}(x+1)} \]

Step 3: Simplify the expression

Now, focus on simplifying the fraction. Notice that \( x^{2}+2x+1 \) can be factored as \( (x+1)^{2} \). Now the expression becomes: \[ \frac{x(x+1)^{2}}{9x^{3}(x+1)} \] Cancel out a factor of \( x+1 \) from both the numerator and the denominator: \[ \frac{x(x+1)}{9x^{3}} \]

Step 4: Cancel the common factor

Notice that the numerator has an \( x \) term which can be canceled with one \( x \) in the denominator: \[ \frac{x+1}{9x^{2}} \] This is the simplified form of the expression after multiplication.

Key Concepts

Fraction Simplification

Understanding how to simplify fractions is a fundamental skill in algebra. The idea behind simplification is to reduce a fraction to its simplest form where the numerator and the denominator have no common factors other than 1. Here's a quick rundown on how to simplify fractions effectively:

• Identify any common factors in the numerator and the denominator.
• Cancel those common factors to make the fraction simpler.
• Continue simplifying until you cannot find any more common factors.

When working with algebraic expressions, pay attention to factoring polynomials which can help reveal common factors. For instance, in the step-by-step solution, notice how: \( \frac{2x^2 - 2x}{2x^2 - 2} \) was simplified to \( \frac{x}{x+1} \) by factoring out and canceling common terms.Remember, simplification helps in making equations and expressions easier to work with and understand.

Factoring Polynomials

Factoring polynomials is all about expressing a polynomial as a product of its factors. When you factor a polynomial, you break it down into simpler components that, when multiplied together, give you the original polynomial. This technique is especially useful in fraction simplification and solving equations.Consider these essential ways to factor polynomials:

• Look for a Greatest Common Factor (GCF): This is the largest factor shared between terms.
• Apply patterns like the difference of squares, perfect square trinomials, or other recognizable forms.

In the example, notice how \( 2x^2 - 2x \) was factored into \( 2x(x-1) \), and \( x^2 - 1 \) was factored as a difference of squares into \( (x-1)(x+1) \).Mastering factoring is key to understanding and simplifying complex algebraic expressions.

Multiplying Fractions

Multiplying fractions in algebra involves multiplying the numerators together and the denominators together to form a new fraction. Here’s a step-by-step guide:

• Write the product as a single fraction.
• Multiply the numerators to get a new numerator.
• Multiply the denominators to get a new denominator.
• Simplify the resulting fraction if possible.

In the problem given, we combined \( \frac{x^2 + 2x + 1}{9x^3} \) and \( \frac{x}{x+1} \) into a single expression \( \frac{(x^2+2x+1)x}{9x^3(x+1)} \). Then simplify by factoring and canceling terms where applicable.Remember, multiplication of fractions can often lead to complex expressions, so it's crucial to simplify to make your final answer more readable and manageable.