In Exercises 15–32, multiply or divide as indicated. (x²−4)/(x²+3x−10) ÷ (x²+5x+6)/(x²+8x+15)

Verified step by step guidance

1. Rewrite the division of fractions as multiplication by the reciprocal. This means flipping the second fraction. The expression becomes: ((x^2−4)/(x^2+3x−10)) * ((x^2+8x+15)/(x^2+5x+6)).
2. Factorize all the polynomials in the numerators and denominators. For example: x^2−4 is a difference of squares and factors as (x−2)(x+2). Similarly, factorize x^2+3x−10, x^2+8x+15, and x^2+5x+6.
3. After factoring, the expression will look like: (((x−2)(x+2))/((x+5)(x−2))) * (((x+3)(x+5))/((x+3)(x+2))).
4. Cancel out any common factors in the numerators and denominators across the multiplication. For instance, (x−2), (x+2), (x+5), and (x+3) may cancel out depending on their positions.
5. Write the simplified expression after canceling the common factors. This will be the final simplified result of the given problem.

Key Concepts

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, recognizing that (x^2−4) can be factored into (x−2)(x+2) helps in simplifying the expression.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This is essential for simplifying rational expressions, as it allows for cancellation of common factors. For instance, the polynomial x^2 + 3x - 10 can be factored into (x + 5)(x - 2), which simplifies the division process.

Division of Rational Expressions

Dividing rational expressions involves multiplying by the reciprocal of the divisor. This means that to divide (x^2−4)/(x^2+3x−10) by (x^2+5x+6)/(x^2+8x+15), you first factor both expressions and then multiply by the reciprocal of the second expression. This process is fundamental in solving complex rational equations.