Questions: Perform the indicated operations and reduce to lowest terms. Assume that no denominator has a value of zero. (x^2-1)/(2x^2+3x+1) ÷ (x^2-1x)/(2x^2+13x+6)

Solution Steps

To solve the given problem, we need to perform the division of two rational expressions and then simplify the result. Here are the steps:

1. Factorize the numerators and denominators of both fractions.
2. Convert the division into multiplication by taking the reciprocal of the second fraction.
3. Simplify the resulting expression by canceling out common factors.

We start by factoring the numerators and denominators of the given rational expressions:

1. The numerator \( x^2 - 1 \) factors to \( (x - 1)(x + 1) \).
2. The denominator \( 2x^2 + 3x + 1 \) factors to \( (x + 1)(2x + 1) \).
3. The numerator \( x^2 - x \) factors to \( x(x - 1) \).
4. The denominator \( 2x^2 + 13x + 6 \) factors to \( (x + 6)(2x + 1) \).

We rewrite the division of the two fractions as follows:

\[ \frac{x^2 - 1}{2x^2 + 3x + 1} \div \frac{x^2 - x}{2x^2 + 13x + 6} = \frac{(x - 1)(x + 1)}{(x + 1)(2x + 1)} \cdot \frac{(x + 6)(2x + 1)}{x(x - 1)} \]

Next, we simplify the expression by canceling out the common factors:

• The factor \( (x - 1) \) cancels from the numerator and denominator.
• The factor \( (2x + 1) \) cancels from the numerator and denominator.
• The factor \( (x + 1) \) cancels from the numerator and denominator.

After canceling, we are left with:

\[ \frac{x + 6}{x} \]

Final Answer