In Exercises 15–32, multiply or divide as indicated. (x2+x)/(x2−4) ÷ (x2−1)/(x2+5x+6)

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

Problem 28

Chapter 1, Problem 28

In Exercises 15–32, multiply or divide as indicated. (x²+x)/(x²−4) ÷ (x²−1)/(x²+5x+6)

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Rewrite the division problem as a multiplication problem by taking the reciprocal of the second fraction. The expression becomes: ((x^2 + x) / (x^2 - 4)) * ((x^2 + 5x + 6) / (x^2 - 1)).

Factorize all the polynomials in the numerators and denominators where possible. For example: x^2 - 4 can be factored as (x - 2)(x + 2), x^2 - 1 can be factored as (x - 1)(x + 1), and x^2 + 5x + 6 can be factored as (x + 2)(x + 3).

Substitute the factored forms into the expression. The new expression becomes: ((x(x + 1)) / ((x - 2)(x + 2))) * (((x + 2)(x + 3)) / ((x - 1)(x + 1))).

Simplify the expression by canceling out any common factors in the numerators and denominators. For example, (x + 2) appears in both the numerator and denominator, so it can be canceled.

After canceling common factors, multiply the remaining terms in the numerators and denominators to obtain the simplified expression. Ensure no further simplifications are possible.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including simplifying, multiplying, and dividing them, is crucial for solving problems involving them. For example, in the given question, both the numerator and denominator consist of polynomial expressions that need to be handled carefully.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This is essential for simplifying rational expressions, especially when performing operations like multiplication and division. In the question, recognizing that x^2 - 4 can be factored into (x - 2)(x + 2) helps in simplifying the expression before performing the division.

Division of Fractions

Dividing fractions involves multiplying by the reciprocal of the divisor. In algebra, this means that to divide one rational expression by another, you multiply the first expression by the reciprocal of the second. This concept is key in the given problem, as it transforms the division into a multiplication problem, allowing for easier simplification and calculation.