Problem 54 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{x^{2}+2 x+1}{x-2} \div \frac{x+1}{2 x-4} $$

Step-by-Step Solution

Answer

The expression simplifies to \(2(x+1)\).

Step 1: Simplify Division into Multiplication

To simplify the division of fractions, change the division into multiplication by inverting the divisor. For the given expression, rewrite it as: \[ \frac{x^{2}+2x+1}{x-2} \times \frac{2x-4}{x+1} \]

Step 2: Factor Expressions

Identify and factor the quadratic expression in the numerator of the first fraction and the expression in the numerator of the second fraction. The expression \(x^2 + 2x + 1\) can be factored as \((x+1)(x+1)\), and \(2x-4\) can be factored as \(2(x-2)\). The expression becomes: \[ \frac{(x+1)(x+1)}{x-2} \times \frac{2(x-2)}{x+1} \]

Step 3: Cancel Common Factors

Cancel the common factors from the numerator and the denominator. The \((x+1)\) and \((x-2)\) factors are common in the expressions. After canceling, you get: \[ \frac{(x+1) \cancel{(x+1)}}{\cancel{x-2}} \times \frac{2\cancel{(x-2)}}{\cancel{(x+1)}} = 2(x+1) \]

Step 4: Final Simplified Expression

The expression is now fully simplified and the answer is: \[ 2(x+1) \]

Key Concepts

Factoring Quadratics

When dealing with quadratic expressions, one of the essential techniques to simplify algebraic fractions is factoring. Factoring quadratics involves rewriting a given quadratic expression into a product of simpler expressions (usually binomials). For instance, the expression \( x^2 + 2x + 1 \) can be factored by identifying perfect squares. Notice it can be rewritten as \((x + 1)(x + 1)\). This is because \((x + 1)^2\) expands to \(x^2 + 2x + 1\) after applying the distributive property.

Why is factoring important? It not only simplifies expressions but also reveals the roots of the equation, thus offering insights into solving quadratic equations. Always start by checking common factor patterns like difference of squares, or perfect square trinomials. Ensure every factorable expression is reduced to its simplest terms, as this will simplify subsequent steps like cancellation in division.

Dividing Fractions

Dividing fractions may initially appear more complex than multiplying them due to the extra step involved. The golden rule of dividing fractions is to "multiply by the reciprocal." In simpler terms, you take the second fraction (the divisor), flip it over, and then change the division sign to multiplication.

For example, in the exercise expression \(\frac{x^{2}+2 x+1}{x-2} \div \frac{x+1}{2 x-4}\), division was turned to multiplication by using the reciprocal of \(\frac{x+1}{2x-4}\), thus becoming \(\frac{2x-4}{x+1}\).

This process ensures that you're dealing with a more straightforward multiplication problem instead of division, making it easier to simplify. Remember, once you have turned division into multiplication, the process becomes about finding common factors and canceling them out.

Multiplication of Fractions

Once we have converted division into multiplication, simplifying fractions becomes a matter of multiplying the numerators together and the denominators together. The essential step here is to multiply fractions directly, unlike addition or subtraction, which requires a common denominator.

Multiplication involves these simple steps:

• Begin by ensuring the fractions are in their simplest form by factoring.
• Multiply the numerators with each other and the denominators with each other.
• Simplify the resulting fraction by canceling any common factors.

In our exercise, after converting division into multiplication, we multiplied \((x + 1)(x + 1)\) by \(2(x - 2)\). Then, divided by \((x - 2)(x + 1)\). This allowed us to cancel out like terms, simplifying the expression to \(2(x + 1)\).

By following these steps, you ensure your final expression is in its simplest form, making it not only correct but also neat and concise.